First, forget buying there unless you plan to retire there. Housing is super-inflated and it takes forever to close on a home. Not trying to trash your dream of eventual home-ownership, but that's the reality. If you've got the money and time, go for it! But otherwise, a good rental company is called Kappel & Kappel (used them for years).
If you want to live close to the base, live in Vacaville. It's a short drive from the base (and really easy if you use the back gate), and has a good mix of nightlife, shopping/restaurants/etc. It's also sorta close to Davis, CA which is a college town and has a little more nightlife to check out.
You can live in Davis, of course, but it's kind of uber-liberal and really crowded because of the college kids. There are some nice things in Davis for sure, but personally I much preferred living in Vacaville and driving over for the eventual Davis outing.
Dixon is a little farming-type community between Vacaville and Davis. It gets crowded once a year with the state fair grounds having events there, but not too bad.
Fairfield (where the base is located) has a few nicer areas, but generally a lot more crime and crappy neighborhoods. You can get lucky and find one of the nicer ones of course, but I wouldn't recommend living anywhere close to North Texas Street, or just Texas st. in general. There's a pretty nice housing area that just came up in recent years off of Air Base Pkwy (which leads directly to the main gate), but I can't remember what it's called. I'm sure the housing office could tell you about that one in particular.
Cordelia is a little suburb of Fairfield, and one of the nicer little areas. It's pretty quiet, and close to Suisun Community College so there's a decent amount of restaurants and such, but not much else.
Suisun is also close to the base, but is really hit-and-miss on the neighborhood quality. Their downtown area used to be really bad, but in the last two years they have renovated it a great deal and it's a little nicer place to visit. (Plus you've gotta check out Bab's Delta Diner for breakfast.) But as far as living there is concerned, it's better than Fairfield but not as safe/entertaining as Vacaville.
Some people live as far out as Sacramento for some reason, it's a hell of a commute from there so I wouldn't really recommend it. There's also Benicia, which is the other direction (going towards San Francisco). It's a pretty little town, but not a quick drive. Then there's a pretty big city called Vallejo...it's kind of like a bigger version of Fairfield: lots of crime and bad areas, although you can get lucky and find a nicer spot once in a while I'm sure.
Aside from crime/ghetto-level/etc., the other bottom line I'd recommend is this: Vacaville, Fairfield, and Suisun are easy drives (more or less). But if you're gonna live anywhere else, do yourself a favor: if you find some gotta-have-it apartment or house somewhere to rent, get up at 0700 while you're in billeting, drive over to it, and see what traffic is like going and coming. Then do it again in the afternoon anytime between 3:30 and 6:00 p.m. Then ask yourself: is this apartment/house really worth dealing with this much traffic every time I go to the base? People usually underestimate how much traffic there is there, even though the big cities of San Francisco and Sacramento are 45-60 miles away.
Good luck...not sure about the squadron that you will work in, but Travis itself is a really nice base, in a lot of ways. Spent 12 years there myself, so it seems like home!