I have been recently looking at finish nailers (15-16ga) and brad/staple nailers (18ga)

I have no clue which I should get.

My primary use will (initially) be crown molding (into drywall and cabinets), shoe molding and possibly some baseboards and picture moldings.

Will an 18 ga brad nailer be sufficient for those and other random around the house joining projects or should I step up and get a 16 ga finish nailer?

An 18 gauge pin nailer should work on crown but no nail will stick in drywall. You need to use a pin long enough to reach the framing behind the drywall. I bought a Bostitch and never regreted it. 18 is on the light side but should be Ok. Stanley Bostitch :  BOS

I was looking at the Bostitch BT200 (5/8 to 2in brad) nailer.

Would you consider this a solid all purpose tool?

Amazon.com: Factory-Reconditioned Bostitch U/BT200K-2 5/8-Inch to 2-Inch 18 Gauge Brad Nailer: Home Improvement

All purpose, no, but a good pin nailer for trim. I own one, they are inexpensive and do the job on trim. However, I also have a 15 and 16 gauge nailer and when doing 1x3 cedar trim switch to these heavier guns.