Get a .22 bolt with an aperture sight. Start at fifty feet. Master the irons. Then put on a decent scope. Practice the steady hold, breath control, and trigger break. A decent scope will let you see your breathing, even your pulse.
I second this. Not taking away from the 10/22, but.... it's not the right tool for this. Nor is a .17 HMR. They are great... but not for this.
Here's the real path, one that is tried and true:
I'd pick up a mid grade "target" .22. Look for an aperature rear, and interchangable blade (globe) front. Put on a reasonably small front aperature, one that is sized to exactly encircle the black of a 50 foot NRA smallbore target (the ones that have ten small black targets arranged around the edges so in a match you shoot one cartridge at each bull and then move on). Get a good sling. And then learn to shoot at 50 feet using this setup. Shoot prone to start. When you can control the trigger-break between heartbeats (when you are slung-in, you will see the sights bounce up/down... up/down... as your heart beats) you will be good enough to move on. You will find that the dual-aperature sights are very precise, and very good. I can have a 12 year old putting every shot into the black *with ease* the first session using this setup. The reason for a bolt action is to slow you down... and they are also easier to find with the correct sights. There are a load of old Remington and Mossberg "target" rifles floating around that are perfect for this. You do not need a high end target rifle. You just need a decent rifle with the correct sights.
After you master prone, work to sitting, then kneeling, and then to offhand. Position is EVERYTHING. Using a sling is MANDATORY. Learn how to build "a vertical column of bones" and the rifle will sit on top of it and only move when your skeleton moves. A GOOD sling on a fair rifle will produce better results than a crappy (or no) sling on the best rifle ever built.
When you are reliably good at 50 feet, scope the rifle and move out to 25 yards. Then 50. Then 100. If you are precise at 50 feet, you will be precise at 100 yards. If you can control position, sights, and trigger at 50 feet you can control them within an equal radius of error at 500 yards. Don't confuse the rifles ability to hit at 200 yards with YOUR ability to hit at 200 yards.
If you cannot hit the X (which is about the diameter of a .22 bullet) on a 50 foot NRA target 9 out of 10 times prone with a .22 using an aperature sight, you are wasting your time trying to shoot further. When you are dissapointed that you put a flier into the "10" and not the "X"... well... you'll be there.
If you cannot shoot a .22 well, you cannot shoot anything else. You are just wasting your time and money.
Sadly all too few are willing to actually put thew work into this. There are no quick shortcuts though.