Accurizing a Remington action is a long and expensive process. This is not something just any gunsmith can do (and do it right). You are looking at several months to over a year to have one of the good guys do the work.
Pillar post bedding is worth it's weight in gold if done right but again, if this is your first, the chances of doing it right are very slim.
What kind of trigger pull do you have. I luv early Remington triggers, but haven't seen many of the newer ones that I didn't have to do a lot of work on or just throw in the trash in put in a Timney. I like a two pound pull, and have my granddaughters at 2 1/2. Just shooting her rifle affects my groups a little just having to deal with that extra 1/2 pound pull. She is used to the 2 1/2 pound and can't shoot mine as good because it's more sensitive than she's used to and has shots she was not really ready for. The trigger must be smooth and crisp or it will affect you're groups.
Next I would have to ask how many rounds have you put through it and did you clean the bore good with break cleaner or a solvent to get the anti-rust out. A new barrel also needs to be cleaned frequently for the first few batches of bullets. It doesn't take much to make 22 cal bullets to give you flyer's.
The next question would have to be how much reloading experience you have. Properly worked brass, the right powder/bullet combination makes a huge difference. I'm working up loads for a 22-250 Tikka Varmint I just bought and with my preped brass, Varget powder and 50gr Nosler Ballistic Tips, I'm shooting bug holes at 100yds. I've got the powder and bullet combo that works good in this rifle and now I will start working it at 400yds to get the final load I will shoot.
I know a lot of this is basic stuff and you may beyound this but this is just some of the stuff that I work on to get the most from a rifle.
Oh yea, and I forgot to mention, you will spend all this time and money to "maybe" get another 1/4" - 3/8" tighter group at 200 yds. Deburring flash hole inside it critical, you must do that. Tightly wrap 0000 steel wool around a bore brush and carefully polish the inside of the necks helps. Chamfering the necks is a must. Uniforming primer pockets helps. Neck turning helps if your brass is more than .0015" out around the neck. Only Neck resizing helps if you have a good, tight chamber.
Another major factor in shooting little bitty groups is the quality of the optics and how good the trigger puller is. 1/2 MOA groups are already good groups, you start seperating the shooters for the rest of the pack when you start getting to that point and smaller, even with a very accurate rifle.