The bottom line is this: what you get out of a trip is proportional to the amount of research you do before you go. What kind of research you do is largely dependent upon what you hope to gain from your travels. For example, if you plan to visit ruins, you will be well rewarded for any historical background you can pick up. While some historical and archaeological sites are beautiful to look at, many are just a pile of stones, unless you have some idea what you're looking at. The ruins of Troy, for example, are some of the least interesting I've ever seen, but anyone who has ever read Homer's Iliad has to appreciate simply being able to stand on the hill where the city once stood. If you can't imagine Achilles chasing Hector around the city walls, then it's nothing but a pile of stones.
If, on the other hand, your primary interest is in meeting new people and exploring new cultures, then it is imperative that you learn the language of the country you are visiting. In Greece and Turkey, I had many self-proclaimed "travelers" (God forbid you call them tourists) criticize me for carrying a camera and taking pictures. "You look like a tourist," they would say, with the implication that the goal was to blend in and be just like one of the locals and "get to know the culture." These same "travelers," however, did not speak a word of either Greek or Turkish, whereas I at least learned enough that I could have simple conversations; more complex ones if I had a dictionary. So, not only did I come back with some wonderful pictures, both of the land and the people I met, but I also learned much more about the people and culture than the self-proclaimed "travel elite" who didn't speak the language. I doubt, for instance, that any of the six or seven people I traveled with to Mt. Nimrod (Nemrut Dagi) had any idea that the town below (Kahta) had a 100% Kurdish population; a very significant detail, if you're familiar with the plight of the Kurds in Turkey, Iran, Iraq and Syria. This fact was only revealed to me after talking with a local family for over an hour, exclusively in Turkish.
Obviously, it is not possible to learn everything. You cannot learn about every painting, every ruin, every castle. You cannot learn every language of every country you may want to visit. Once again, you will be limited by time and resources. It is not necessary to know everything, just as it is not necessary to visit every piece of ancient marble on a trip to Greece (believe me, I tried). I don't speak a word of Dutch, but I still enjoyed the Reijks museum in Holland. I don't know the history of the Medici family in Florence, but the Medici chapel was still an incredibly beautiful piece of architecture. Try to prioritize, learn what you can, and make the best of the time and money that you have. The more you learn, the better, keeping this in mind: if reading about something doesn't interest you, seeing it probably won't, either.
You could spend your entire life just studying a single town, or even a single building in a single town. The resources available are without number. What resources you use should be dictated by what kind of information you are interested in, and what kind of depth you hope to gain. If all you want is basic geography and culture on a very superficial level, an encyclopedia will usually do the job. You'll get a couple paragraphs on each country, and what you get out of it will follow accordingly. Classical history buffs may find textbooks particularly useful, or even taking a class on a specific region. The World Wide Web (internet) can be either indispensable or completely useless. (One of the problems with the net is that there are few if any "editors," and thus anybody with an opinion (like me!) can pretend to be an expert. Taken with a grain of salt, however, the net can be an invaluable resource.
If you decide to use the net, check out my links to other sites. I recommend starting with the U.S. State Department's website, http://travel.state.gov/. They have a wealth of information, most of which is generally considered reliable. When all else fails, use a search engine (http://lycos.com or http://yahoo.com).
Other resources include magazines (National Geographic, TNT), travel guides (Lonely Planet, Rough Guide, Let's Go), documentary videos and textbooks, among others.
This is actually a subject of much debate among travelers, so I can't offer a definitive answer. I have heard it said that some of the guides do well for some countries, others better for other countries. In my experience, the Lonely Planet guidebooks are excellent. I have compared them with other guidebooks and I continue to use Lonely Planet. (I was pretty unimpressed with their website, however.) I believe there is a series for the Mediterranean (I know they have one for Greece) called something like the Blue Guide or the Blue Book, which is extremely thorough but is fairly dry reading.