Playing the roots in the groove is fine, but consider that for a lot of songs there is a particular bass line or pattern of notes that forms a "hook" for the song; something that detracts from the song if not done, or at least makes it plainer than the artist intended. If you're covering a hit, there's a reason it was a hit, just maybe the original bass line had a lot to do with it. Why not try to mimic that once you have the root notes down pat for the chord progression?
I'd say to listen intently to the tracks you're covering, focus on the bass, and try to pick up on some of the things the bass player is doing. You can use your computer or phone software to EQ the bass part up a little louder, and some software allows you to slow down the tempo of an .mp3 without impacting the pitch. That's a great way to figure out those things and add them to your performance when you can duplicate them well.
Once you figure that out, you're free to apply what you've learned to other songs IF it feels appropriate. Try it at rehearsal and see what your band mates think. They may want you to keep it simple, or they may encourage you to explore further.
Edit: Picking up some theory along the way certainly can help, but don't get lost in it. It is merely a tool to play better, and to understand WHY you're playing extra stuff; learn BOTH along the way. Learn what a fifth means, sure, but more importantly learn how it sounds and that's why you want to add it.
Something else to consider; with a 5-piece band you have to leave a lot of space for everyone else, usually simple bass lines are all that is needed, depending on the piece. But if you're playing with just a guitarist and drummer, then you may want to think about adding things you learn to help fill in SOME of the space, not every available nook and cranny of course.
The last thing I'll say is make it FUN. I didn't pick up a bass until I was 50; now I'm 62 and have been playing out for 9 years with several different bands, playing many kinds of music. It hasn't always been easy, but it's worth the effort, and rewarding. The great trio I'm in now will gig 25+ times this year, and this is about the most fun I've ever had! Just keep working on it, with your obvious drive to improve, you're going to have a blast!