Given that, imagine how much more difficult it is to probe the beliefs of early Christians. The ancient mind was as variable and as subject to individual circumstance as our own, and exactly the same kind of factionalism existed between early Christian groups as exists presently between modern Christian denominations. The titular question is rendered even more difficult to answer because systematic theology is largely absent from the earliest Christian writings (Romans and Hebrews perhaps come the closest to being so), which means that our understanding of the earliest beliefs must be deduced entirely from the passing references to heaven and hell that exist in the surviving texts, which obviously leaves a lot of room for ambiguity and guesswork. So, in addition to the kind of definitional problems that preclude us from answering the question about what modern Christians believe about heaven, we can't know for sure what the first Christians believed about heaven and hell because they never actually bothered to tell us. (Although this realisation in itself may be significant: the peripheral treatment of heaven and hell in the NT may tell us that it was not an issue of great interest to the earliest Christians, at least not in comparison to questions of the Law, eschatology, Christology and so on.)
Having said that, we can probably say with some certainty what the earliest Christians didn't believe about heaven and hell and that might be the best place to start. The modern idea that Christians go to heaven by virtue of their faith in Christ is a Protestant idea that postdates Christian origins by some 1500 years. It comes from Martin Luther's concept of sola fide ("by faith alone"), which was a reaction to the then prevailing Catholic view that entry to heaven was governed by performance of the Sacraments, which themselves required the mediation of priests. In modern times this concept of sola fide has been taken to a particular extreme by certain Protestant fundamentalists who appear to believe that entry into heaven depends solely on whether or not one assents to the metaphysical claim of Jesus' divinity. I think the best that we can possibly say about such a theology is that it's a lazy one, but in any case it certainly isn't Biblical. In fact most of the ideas affecting modern Christian views on heaven and hell are post-Biblical, inspired largely by later systematic theology and the vivid medieval imagery of Dante and others. The Biblical references are - as I have already mentioned - sparse, inconsistent and often confusing.
The first Christians were obviously influenced first and foremost by Judaism. The Israelites of the OT period didn't really have a concept of an afterlife, although they did, of course, believe in a "heaven". This heaven, though, was literally God's home - a great Kingdom existing above the firmament in the sky - which he shared with the angels (and the other Gods?) in something resembling a great city or palace. This was a place completely closed to human entry, with the exception of the two prophets Elijah and Enoch, who apparently ascended there without ever dying. There was also belief in a shadowy netherworld referred to as "Sheol" that one may have found oneself in after death, but it was apparently not seen (uniformly) as a place of either reward or punishment, and nor is there any indication that existence there would be a permanent or eternal one. In this respect, it shared a great deal in common with other near-Eastern views of the afterlife from the same period. So how did we get from such beliefs about the finality of death to the apparently common belief in the NT that one could - under certain conditions - enter God's Kingdom and be granted eternal life?
To answer this question, we need to look at the theological developments in Judaism during the so-called "inter-testamental" period, roughly 200 BCE - 70 CE. For reasons that aren't entirely clear (though it may be related to the perceived "injustice" of the righteous dying during the Maccabean revolution before their vision of a new Kingdom was realised), the Jewish belief in the resurrection of the dead became common (though by no means universal) during the inter-testamental period, and it is usually found interwoven with the eschatologies that also became common during this time. With the restoration of the New Jerusalem, the dead would be raised to share in its glory, not in some higher spiritual plane (it seems) but right here on Earth. More to the point, the dead would be physically raised - that is, they would be resurrected to walk the Earth in physical bodies much like their old ones. Did such beliefs concerning the afterlife carry-over into early Christianity, or did expectations change in the post-easter period?
Responding to the Corinthians, who were apparently concerned that the dead would not get to see the coming Kingdom of God, Paul assures them:
1 Corinthians 15:12-20:
Now if Christ is proclaimed as raised from the dead, how can some of you say there is no resurrection of the dead? If there is no resurrection of the dead, then Christ has not been raised; and if Christ has not been raised, then our proclamation has been in vain and your faith has been in vain. We are even found to be misrepresenting God, because we testified of God that he raised Christ—whom he did not raise if it is true that the dead are not raised. For if the dead are not raised, then Christ has not been raised. If Christ has not been raised, your faith is futile and you are still in your sins. Then those also who have died in Christ have perished. If for this life only we have hoped in Christ, we are of all people most to be pitied.
But in fact Christ has been raised from the dead, the first fruits of those who have died.
Note firstly that this passage (and others in Paul's letters) refer to this resurrection of the dead as a future event, as something that will happen at a singular and definitive moment in history (namely when "the trumpet will sound" 1 Cor 15:52), and also that such happenings were not yet current at the time of writing (though the resurrection of Jesus was the sign that such an event was imminent). The idea of a resurrected soul at the moment of death (which is what the current Christian understanding of the afterlife amounts to) has not yet entered the zeitgeist of Christianity, and won't for quite some time yet. In any case, quite how we should understand an expression like "resurrection of the dead" in Paul's writings is open to debate, but he wrote extensively enough on the subject for us to hazard some timid assumptions. The first and most important is to explore what the nature of this "resurrection" really was, and if it differed in any substantial way from the Jewish beliefs that preceded it.
Paul says to the Corinthians that the dead will be raised into "spiritual bodies" (1 Cor 15:42-44), but a contrast with Paul's other writings tells us that such a belief was not necessarily normative in Paul's writings, let alone among other early Christians. It is true that Paul believed the resurrected body would not be identical to the earthly body (i.e. it will be "transformed" at this moment - Phil. 3:21), but he also makes it quite clear that the resurrection will involve the "mortal body" (Rom 8:11). This is consistent with his wider belief that the resurrection of Jesus marked a fundamental turning point in history, and that everything to come would be fundamentally "renewed" through God's grace (e.g. 2 Cor 4:16-17). So although we repeatedly find tension in Paul's writings between "the spirit" (pneuma) and "the flesh" (sarx), we shouldn't assume that Paul held a kind of Platonic dualism, where the immortal "spirit" and the perishable "body" were held as strictly demarcated entities. Paul was moved to write the above passage in response to the skepticism of the Corinthians ("how can some of you say there is no resurrection of the dead?") who shared more typically Greek conceptions of immortality - if Paul believed that Greek philosophy was right to scorn the prospect of bodily resurrection in favour of the immortality of "the spirit" then he would have indicated so here. That this passage exists is proof enough that Paul (and so, presumably, other early Christians) generally believed in a bodily resurrection and were motivated to defend it against the charges of its more stubbornly Hellenic adherents.
This idea of a "physical" resurrection is more clearly demonstrated among the writing that would later become the New Testament corpus. The empty-tomb tradition and the subsequent appearence of Jesus in his physical, still wounded body (cf. Jn 20:25-28) seems to indicate as much, as well as other similar scattered passages that clearly refer to the physical resurrection of the dead in the NT (e.g. Mt 27:52-53). The upshot of all this is that while in late-2nd Temple Judaism and early Christianity we have unequivocal signs of a belief in an afterlife, it's still not one recognisable to modern Christian theology. Souls aren't floating gently up to heaven in this scenario, rather the dead are expected to literally be revived to once more walk the Earth.
Connected to the belief in the resurrection of the dead was the belief in some future involvement of God here on Earth that would radically and eternally transform it to make such a development possible. Such beliefs can only be termed "eschatological" and it was such eschatological beliefs that informed new understandings of heaven among the early Christians. As I have said in earlier posts, Jesus (and his later followers) seem to have believed in a future state where the Kingdom of God would be literally implemented on Earth and it was in this future reality (still part of the world, though a drastically transformed one) that eternal life might be possible, though - again - the mechanics and terminology of this event seem a little confused. Can we so easily equate "Kingdom of God" with "Heaven" with "Kingdom of Heaven"? Will heaven come down to us (the Synoptics?) or will we be raised to heaven (Paul and Hebrews)? Such questions defy easy answers, and it makes it difficult to talk about what the precise beliefs of the first Christians were with respect to heaven and life after death with much confidence.
A further confusing factor is the influence of Hellenistic philosophy and how it may have influenced early beliefs over and against those of Judaism. Paul, the earliest surviving Christian writer, seems to have been influenced by Hellanistic philosophy on these matters to at least some extent (e.g. his reference to a "third heaven" - 2 Cor 12:2 - appears to be derived from Platonic schemata of the heavenly realm) but it's not clear what significance we can draw from such observations. As mentioned earlier, Paul plainly doesn't subscribe to the wider Greek philosophy concerning the eternal nature of "the soul". One important Hellanistic influence on early Christianity, however, seems to be the notion of hades - the Greek underworld - most commonly translated as "Hell". This, obviously, was a fiery place of torment where the souls of the unrighteous were fated to go, though I'm not sure if it's clear how or why such a concept was incorporated so early into the Christian tradition (it is regularly mentioned in the Synoptic Gospels though not - perhaps tellingly - in either Paul's letters or in gJohn). Can such a belief be traced back to Jesus?
Probably not directly. Familiarity with Hades and a willingness to incorporate it into one's world-view would have required a familiarity with Greek philosophy that probably would not have been accessible to a marginal Jewish peasant like Jesus. Likely it was something invoked by the Synoptic Gospel authors to better illustrate to their gentile readers a concept that Jesus likely did talk about in his time - that of Gehenna (e.g. Mk 9:43-48 etc.). Now exactly how this term was understood in Jesus' day is still being debated, but it bears emphasising that it was a real location, situated in a valley outside the boundaries of Jerusalem, that was associated with Pagan sacrifice and the burning of dead criminals (i.e. those who had died in sin). If we presume that Jesus had similar understanding of this place, then we can (with some caution) speculate that Jesus believed that this was the destination for those who were not fit to inherit the Kindgom of God. In other words when the Kindgom of God was to be implemented on Earth (in the form of a "New Jerusalem"? cf. Rev 21:2) those who had lived up to God's expectations would be admitted, but those who didn't would be consigned to Gehenna to burn. However, even if such an interpretation portrays accurately the beliefs of Jesus, we should note that Gehanna should be treated as a different place to that of Hades, and that it is misleading for modern English Bible translations to translate both terms as "hell". Our current understanding of hell would almost certainly seem foreign to Jesus and other Jews of his day.
So, to recap, modern Christian understandings of heaven, hell and the afterlife in general are predominantly later developments, though they do have their seeds in the theology of the first century. The ambiguity of terms, mixed expectations of the authors and paucity of references in the NT, however, left plenty of flexibiltiy for later theologians to both re-interpret and embellish these early accounts with some degree of freedom, which they clearly have taken advantage of. If you want some clear insights into the nature of heaven, hell and the afterlife, though, then the Bible probably won't help you much.