First shoot the rifle on a solid rest 5 (or 10) shot groups at 50 yards. Decide exactly how the gun shoots now and keep this in mind as you move forward. You will always have a base to go back to.
Second decide what purpose you want the rifle for target, hunting, experimentation, plinking, accuracy buff ect. If the rifle is already accurate and functional to serve your purpose improving it may not IMPROVE it. A rifle may serve more than one purpose but the more reasons you change a rifle for the less any one of those is improved by the change.
The 77/22 has many mods you can get to improve the rifle without doing any gunsmithing. First Midsouth Shooter's has a trigger sear and spring which will just drop in (about $30) if you are not afraid to take the trigger apart and reassemble it with the new parts. More expensive parts for the trigger also exist if you want a really target trigger. As many drop in barrels exist as for the 10/22 (almost) but a new barrel may not help much if any problem listed below exist.
After you get the trigger to where you like it you need to check the rifle for function. The 77/22 has a few areas that have to be right for best accuracy. Riles with these problems generally must be fixed before they will tune.
1. Lug noncontact, you need to check if both locking lugs (on the bolt) are contacting the reciever when the bolt is forward and cocked (safety NOT LOADED) if the rifle has a scope it is best to remove it to get a good look at the lugs. The better the lugs fit the better the CHANCE for accuracy and this is why some lap the lugs fo perfect fit.
2. Barrel Cant, the barrel reciever contact on some rifles (even new ones) is not very solid. This lets the barrel sag more than it should when the barrel lock is tighened a lot of pressure is exerted on the barrel and stock. The common methods to remedy this are to send it to be fitted (by gunsmith) or cement the barrel/reciever. Cementing is not the best proceedure if one is not skilled because the proper cement should used (if you need to take the barrel off improper cement won't let you) and the barrel needs to be held in correct position while the cement dries.
3. Forward sling screw in contact with the barrel. This doesn't happen often but I've seen it. You can simply file the the sling screw to correct this if it occurs.
Lastly for anyone who wants to make a rimfire shoot must know something about sights (scopes mostly). Most 3x9 scope are not very good for a rimfire as they have the wrong paralllax setting. I will not say what parallax is but it is important if shooting small targets. Needless to say however the rimfire should be fitted with a scope that has the proper parallax setting or a scope that can be adjusted to counter parallax error. These adjustable scopes are called adjustable objective (A/O) types.
Perhaps this info will help someone with a 77/22 perhaps not but KEEP SHOOTING ANYWAY.