Congrats!! Welcome to the addiction
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For a tuneup, I think really all you need is to have the rear derailuer adjusted. That runs $10 here, at the expensive shop, so I think it shouldn't be too bad for you. Pick up a bottle of chain lube, and ask them their opinon on the brakes. Then work on the boss to let you get something else
Really though, I started riding a 10 year old Raleigh MTB that was barely rideable, but I had no idea. I just rode and rode, when something broke I replaced it (which was often). Once I hit the $100 I decided that it was time, and bought a $350 bike that has been the second most reliable I've owned.
For now, just ride and work up to what you would like.