I have an ER Shaw 10/22 heavy barrel, a factory laminate stock and factory receiver w/ factory internals.

What should I upgrade?

Was thinking trigger but not sure what else. I'm looking for decent accuracy.

Suggestions on build as well as what brand parts please.

ER Shaw barrels are no slouch and a heavy barrel eliminates a lot of issues with any pressure on the forearm throwing shots, so you're GTG there.  

My advice would be an upgraded trigger.  A competition trigger and sear from Power Custom can be had for under $100.  Ruger's drop-in trigger upgrade is supposed to be pretty nice, too.  
If you want to go big, buy a trigger from KIDD or Volquartsen.  It really depends on your budget and what you're willing to throw at a 22LR.  
Another suggestion would be to buy an upgraded extractor, but typically that's only a benefit if you're having extraction issues because of a match barrel with a tight chamber.  

What optics are on the rifle?  

Thanks for the suggestions.

Optics is another issue. I am not sure what I am going to put on it.
I am open to suggestions on that as well.

http://www.sportsmansguide.com/product/index/sniper-6-24x50mm-tactical-rifle-scope-matte-black?a=990001


I put one of these scopes on my 10/22 on a lark. It is great! You can really wring some good groups out of a 22rf with one of these. I put the Kidd trigger kit, poly buffer, and extended mag release lever on my rifle. I can literally shoot tacks using the Rimfire Central Forum tackdriver targets at 25 and 50 yards. Yes I know it is way overglassed, but who gives a damn. It works very well for an inexpensive solution to the question of what optic.

Or, you could blow a years ammo budget and buy a Schmidt and Bender or some such scope................

Upgrade the trigger for sure. I would personally do the KIDD single stage drop in unit. If you want to save a few bucks, the KIDD trigger kit for just over $100 will greatly improve the trigger and is well worth the money. If you are looking to save a buck, the BX trigger is a huge improvement over the factory and drops right in.