I used to play a lot of bass for various gropus, and I still do all the bass work on my solo stuff. I really love playing bass - it's a beautiful instrument. One thing I'd say is that keep in mind that it is a different instrument, with a different role in the band. It's not really a 4-string guitar tuned a couple octaves lower. A lot of it, in my 2c opinion, is knowing when to just hang on the root and when to add melody. Putting a nice melodic line in just the right place can really bring a song to life. Conversely, a bassline that's too busy and "guitar-like" can sap the groove out a song. Like Big E said, you might want to just hang on the root or riff and lock onto the drummer while you're getting a feel for it. One other thing - you'd better know the song structure inside and out! Guitarists can flutter about on top of the song, or even stop playing entirely and nobody will really notice (especially if there are two guitars). If the bass flubs, it's really noticeable because it's the foundation of the song. Learn to count the measures while playing!
Like any instrument: listen to bassists you like and try to emulate them. Play along and figure out what they're doing, when and why.
A mate of mine once made a comment about guitarists looking at a fretboard in "shapes" and bassists see it in "runs and patterns"...
I don't know about that. I can see where he's coming from, but it's all the same 12 notes, the same scales, and the same chords whether you think in shapes, patterns, runs, notation, or whatever. The same patterns and runs work on guitar and bass since the tuning is the same, it's just a difference in what and how you're playing on each instrument.
Regardless, good luck and have fun. I always have a blast playing bass.