I keep reading and hearing about all these forward-thinking research projects that were being pursued in Nazi Germany, many of which were picked up by the Soviets and the United States. What was driving all this advanced research?
The Nazis believed that new technology was going to lead to new breakthroughs in war, among other things. They learned this lesson, as did everyone else, during World War I, where innovations in artillery, machine guns, flight, tanks, submarines, and chemical warfare seemed to matter a hell of a lot more than tactics, generals, morale, and things like that.
The Nazis embraced this fairly early on, but they were not unique in this approach. The United Kingdom and the United States both invested heavily in wartime science and technology. The UK started doing this around the same time as the Germans, with an eye on future war. The US started a little later in the game — they got their act together around 1940-1941 — but there were those in the country who were lobbying for this even earlier.
What makes the Nazis look "forward-thinking" is in fact that they bet on projects that turned out to be beyond their capabilities to fully exploit for the war. Rockets are a great example. The work on the V-2 in particular was way ahead of its time. So far ahead of its time that it had really no effect on the war at all; the furthest they could get it was not far enough to matter very much.
The UK invested in technologies very specific to their aims in defending against a German invasion — radar and cryptography, for example. On radar they realized very early on that they couldn't get it completely up and running in time to be useful, so they handed it off (literally: they brought a cavity magnetron across the Atlantic) to the United States and asked them to do it.
Similarly they realized that they didn't have the capability to bring atomic bomb work to any fruition quickly and encouraged the United States to do it as well, which eventually jump-started the serious US program.
The Soviets' interests in new technologies was at a somewhat different "stage" than the US, UK, and Germany because they were still involved in the process of rapid industrialization at this point. The kinds of things they focused on were big capital projects (e.g. the Dnieper Dam), electrification, better agricultural output, and etc. During the Stalin years, just before the war, they were also wracked by ideological factionalism which led all of their sciences (not just biology, which is the most famous) to undergo all sorts of contortions to avoid purges. They were caught flat-footed once World War II began and their difficulties against the Germans didn't exactly free them up for lots of R&D, but they got their act together over time, and certainly by the end of World War II had realized that big research was the way to go. They saw themselves as playing a lot of catch-up in the Cold War for this reason.
The US looked at all manner of technologies. Radar and the atomic bomb are the flashiest outputs of this, but they also developed some incredible, important, and practical technologies like long-range heavy bombers (the B-29 was pretty impressive for its time and gave the US a distinct advantage in the Cold War), the proximity fuze (which allowed even misses to be hits), sonar (which did gang-busters work against submarines), napalm/incendiary bombing (which proved ruinous to the Japanese when combined with the B-29), and a whole host of other more minor technologies. Almost all of these paid out hugely in terms of investment — they actually mattered with regards to the outcome of the war.
It's true that the US and the USSR eagerly took in some of the work of the Germans, notably rocketry, which neither had invested much in previously, and both of which realized that, when coupled with a nuclear warhead, would change the nature of everything. But as a wartime investment, rocketry was a real flop for the Nazis. It produced some good propaganda, but it did nothing in terms of helping their war aims.
One nice little volume on the Nazi party's own approach to technology — which was a key plank in how they saw their reform of the German state — is Jeffrey Herf's Reactionary Modernism: Technology, Culture and Politics in Weimar and the Third Reich. What Herf is trying to explain, in part, is why the Germans could be simultaneously embracing of all of these very anti-modern notions (of race and blood, of bread and Volk, and often be against many aspects of modern science) and yet still be super enthusiastic about making technology.