I've been thinking about trying to learn either Finnish or Swedish, just because I really want to travel to the region someday. Any pointers?
Well, for starters the two languages are completely different. Swedish, like English, is a Germanic language, and due to the contact between the Norsemen and the Anglo-Saxons, there's quite significant close similarities. Finnish, on the other hand, is a Finno-Ugric language, related to Hungarian and the Baltic and Sami languages - as well as several community languages spoken throughout Russia.
If by 'region' you meant Scandinavia, I would reluctantly advise you to try your hand at Swedish. Swedish, Danish, Icelandic and the two varieties of Norwegian (bokmål and nynorsk) are quite close to each other, although not completely mutually intelligible.
if, however, you should come to Finland, then by all means, study Finnish ^_^ Just a cautionary advice, though: Finnish is often ranked among the most difficult languages to master, right after Mandarin Chinese. Fifteen different cases alone tend to scare people away.
Generally speaking, however, you should be able to get by with English in all of the Nordic countries. Most people have been studying English as their first foreign language, and at least the younger generations are extremely fluent in it (at least in Finland).
Fun Fact: Finnish was one of the languages that J.R.R. Tolkien used when developing Quenya. The pronunciation of Finnish and Quenya are quite similar, and there's a lot of words that are either directly from Finnish or can be easily traced back to the Finnish base word.