These suggestions are based on my experience with my two Ruger 10/22’s.
Trigger and sights. Start here.
I don’t know what your intended use for the rifle is: plinking, hunting, or precision. I recommend upgrading your trigger and sights/optics, then testing your rifle with a goodly amount (500 rounds or so) of your preferred ammo. Many stock 10/22s are plenty accurate out of the box. But if yours is not up to your expectations, a barrel swap is always an option. I've never done that, so I'll defer to previous comments for advice in that arena.
TRIGGER
There are lots of triggers out there. Ranging from 75 or 80 bucks on upwards towards 300.
I chose not to break the bank and went with the Ruger BX trigger. I’m happy with it and don’t feel like I “settled” at all.
Mine dropped the pull weight from 5-6 lbs to 2.5-3 lbs . It's a factory drop-in that's improved my accuracy. I ordered mine on eBay for around $80. You can also find them at Midway for $77 plus shipping, tax, etc. 10 min drop-in; no tools beyond a punch.
https://www.midwayusa.com/product/1016234618?pid=534496
While you’re in there with the trigger, add an auto-bolt release. It’s another piece that drops in to the trigger group you’re installing. You can currently get a two-pack (because 10/22’s are like potato chips, don’t ‘cha know?) on eBay for 13 bucks. Well worth it. No futzing with that little lever at the trigger guard every time you want to drop the bolt closed. Just a tug on the charging handle and you’re back at work.
https://www.ebay.com/sch/i.html?_nkw=10/22+auto+bolt+release&_sacat=0&_from=R40&_trksid=p4550459.m570.l1311&_odkw=bx+trigger
SIGHT UPGRADES
I’m 67. The factory sights are a non-starter for my
old “experienced” eyes.
Iron Sights:
I was setting my gun up to take the Appleseed course and a brand of iron sights from Tech Sights were highly recommended. After using them for years, I recommend them as well.
Get the TSR 200 model. Again, eBay is your friend.
https://tinyurl.com/tsr200
Optics:
The world is truly your oyster here. Don’t go $50 scope, but no need to go $500 either, IMHO.
I am happy with my Burris Droptine 3-9 x 40. $125-130 on Amazon.
https://tinyurl.com/Burris-Droptine
Also, depending on your intended use, nothing wrong with a fixed 4x scope. I currently run a 4x Nikon Rimfire (sadly out of production now) on my 10/22. I swap it out sometime for the Burris, depending on my mood/needs.
I haven’t shopped for a fixed power in years, so you’ll need to do your homework on this one. They’re not everyone’s cup of tea.
Hope this has helped some. Welcome to the rabbit-hole!