I just found out I'll be moving to Michigan for a job this summer. While in Michigan I'm really hoping to get a start into playing music live. I'm thinking because of the new environment I won't be embarrassed about people's reactions like I would be back home, plus I'll be headed back home sometime in August anyway.

I'm not going to worrying about trying to throw a band together since my time there is pretty short, but I'm wanting to start playing some solo shows. But truth is my stage experience is very little. I play on stage with a church group once and I played a song with a rock band once at a friends birthday party (but the whole crowd was intoxicated so they probably don't know what I played)...and that's it. I assume it would be best for me to hit up some open mic nights first? And if it goes well maybe work on getting a small gig somewhere? Also, assumedly they provide the mic and PA system for open mic night, but will the mic pickup on my guitar playing? Or will it probably only catch my singing? My guitar doesn't have an amp plugin btw. I was hoping to switch between guitar and banjo on stage. Any tips would be appreciated!
Normally you should have two mics, one for the acoustic and one, higher up, for the vocals.
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I would be surprised if moving to a new environment makes you less shy about performing. There's no need to put off performing at open-mics until you move to Michigan. You should give it a go as soon as you're confident that you have enough material of a good standard to play. You should consider going along to a few open-mic nights just to watch, it might help you build your confidence.