I'm guessing you got a new model 10/22 and not one from the 90's and older? I say this because the older ones had aluminum alloy trigger frameworks, while the new ones use plastic.
I'd start out with a BX trigger system that is a complete system and just a simple drop-in and replacement of the 2 receiver holding pins to give a better trigger response over the factory.
Your 10/22, if new, should have a threaded on receiver type barrel, so simple swaps of barrels is now pretty much out of the question. This is one area where the older 10/22 had a good advantage in that one could choose what aftermarket barrel maker they liked or reviewed in the choice for accuracy, customization, barrel length and if they wanted iron sights or not for their swaps.
You'll find that if you want to change from that walnut-stained hardwood stock to something a bit fancier, there's Richard's Microstocks where you can get a new stock in real walnut or choose one of their fancy laminate color stocks that will WOW your friends.
There's also the selection of scope brands from cheap to ultra expensive that one can go with, if you so choose, as I believe even the new 10/22's are drilled/tapped for base plates.
Best thought I have is find what ammo the rifle likes best for the accuracy you seek, so buy some boxes from different ammo manufacturers and do your own tests to let the rifle tell you what it likes the best.
The 10/22 is the original LEGO platform next to the AR15 in what a person can do or wants to spend.
