As said, you don't need fluency in another language for linguistics (though some programs require it), nor do you need it for computational linguistics. I've never heard of any major requiring you to know any knowledge of a foreign language before you declare, but some require you to become proficient on at least one other language as a degree requirement. Remember, though, that linguistics is not about the study of any one particular language or languages, but about the science of language: the structure behind it, what makes it work, how it changes, how languages relate to one another, how language is integrated into society and into the mind, etc. Because it's a science, there is formalism; of all the social sciences and the humanities (different universities classify linguistics differently), it's probably the most "techy," but you can get around much of the "techiness" if you'd like (i.e. taking more sociolinguistics, psycholinguistics, etc. courses).
Linguists do study abroad, though I'm not sure whether they do more so than other majors. I can tell you that linguistic fieldwork is pertinent these days (nearly half of today's languages will be gone by the end of the century), and that does require significant traveling.
Here's a good article on why one would major in linguistics and what one can do with it: LSA: About Linguistics
Check out linguistlist.org to get some insight into the community behind linguistics; it's very active, and has lots of job listings, conference calls, papers, books, etc.
Would I have to double major in computer science and linguistics to get into computational linguistics?
No. You could major in one and take a bunch of courses in the other. CL is a flexible field.
I had originally (for a few years) intended on majoring in linguistics and computer science for the goal of computational linguistics, and while that goal has not changed, I'm not majoring in either, but instead in a field (symbolic systems) that combines the two with lots of math and logic, as well as psychology and philosophy. And majors in this area do plenty of work in CL.