OP, definitely go to the track where you'll have an instructor riding shotgun. It's not especially cheap, particularly if you have to buy dedicated track insurance because your normal policy won't cover it (mine does, one of many reasons I'm with Amica), but IMHO it's one of the best fun-per-dollar experiences available.
As for mods, I would say if you've never been to the track before, you could probably track this car bone stock and be just fine; just get your car inspected beforehand to make sure everything's in working order and you have sufficient tire tread, brake pad thickness, fluid levels and "newness", etc. If you keep with it and get better, brake fluid will be your first issue (spongy pedal), which can be fixed by running something like ATE Super Blue or Type 200 (same stuff, just different colors), and then brake pads will probably be next to deal with fade. Then much later on, suspension mods might be considered to address understeer (once you can handle neutral handling), then OPTIONALLY tires if you want to run slicks and are willing to pay for them, in which case you'd probably want a BBK to deal with the extra brake heat generated from all that extra grip, and then FINALLY engine mods MAYBE -- but all of that would be much later. Notice how suspension, BBKs, and engine mods are LAST in that list even though they're the mods a lot of people jump straight for, skipping over the less sexy items? Yeah, that's how you separate real drivers from poseurs. The first three mods of any real driver and the ones that make the largest speed difference by far are the driver, the driver, and the driver.
As you get better, the habit does indeed get more expensive since brakes and tires wear faster, fuel economy decreases, you start thinking about more expensive mods, and of course you'll want to go more often. But even if you don't become a regular, it's a great thing to do in this car even once because the M3 can't be fully appreciated when confined to public roads, especially US roads. The track is what it was built for, after all!
As others have said, just leave your ego at the track entrance, keep DSC on to start, listen to what your instructor is telling you, and don't hesitate to let people pass you so you can focus on what you're doing rather than what's behind you. Accidents under those conditions are highly unlikely, but a fun time is virtually guaranteed. Oh, and the more you drive on the track, the more mellow you'll be on the street -- then you'll stop thinking about doing stupid things like treating a canyon road as a track and being paranoid about cops.
