One reason a research strategy can become so powerful is the way research tends to gain its own momentum. Because researching allows you to gain access to better, more advanced technologies, which in turn improve your empire in numerous ways, spending a lot of money in research can turn you into a juggernaut. Eventually, your technology will be giving you so many bonuses that no one else can hope to compete. But to reach that point, and reach it quickly, you'll need to spend wisely.
The most obvious place to spend your research dollars is in research technologies. That said, it is easy to mistakenly neglect research technology, because it is time consuming to research and does not lead to an immediate benefit. It may seem difficult to justify researching an upgrade to your labs when it is possible to research two or three weapon upgrades in a similar span of time, but take the plunge. It will pay off massively. With better labs, your research capabilities will gradually increase over that of your foes. In ten turns, the difference is difficult to see. But in one hundred turns, the difference is massive.
Another place to spend your research dollars is on production technology, particularly those which give you an empire-wide production bonus. Not only will your factories be able to create warships and social structures more quickly, but you'll also improve your overall efficiency. Although it is never stated by the game's tutorials or tips, bonus production actually costs less money than regular production.
One area you should probably not spend to much time focusing on is diplomacy, provided you already have an edge. Becoming a skilled diplomat through research is a good idea, but if you become overly skilled, you may find yourself with little to use your diplomacy skills on. Other races are not going to part with planets, or with their favorite technology.