TK0218 my advice is the same as Willie. Learn to walk first, the 22 is not a toy and small bore shooting is not easy but once you master that it will carry over to your centerfire shooting. Bolt action, aperture sight, single shot or magazine fed your choice. Target barrel on the gun will reduce some of the climbing problem. Try a few different types of ammo to see which one your particular gun likes. 40 gr solids will give you the most consistent performance all the way out to 200 yards. You do not need Mr Hyper super uber velocity, standard velocity target ammo will do just fine. All you really need is to be able to shoot thumbnail size groups at 50' free style, thumbnail size groups from sitting or prone at 50 yards and under 2" at 100 yards from a rest and you will be good to go.
Per your M1 problems, yes the barrel will warm up, the 30-06 is a nice little heater but your climbing should be minimal. Get a good gunsmith familiar with the M1 Garand to go over it and correct any problems, ie gas port plugged, un even bedding, bad crown, bad ammo, improperly slung up, could be 30 different reasons.
I won't name a particular brand or model, most will shoot better than you and you need to find one that fits you. Makes you grin when you pick it up and hold it in the offhand position. Look for something like a Savage, Remington, CZ, Kimber etc. with a heavy barrel. New or used it won't make much difference as long as its in good shape.
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