For a grand I would go with a basic Ruger 10/22 Carbine 22lr and a SKS
CSSA Member
Another vote for a .22LR bolt action. The ammo is cheap and if you are worried about recoil, there is really none. You can learn good fundamentals of shooting and with a decent scope, you would be encouraged with results which will just prod you to continue shooting. I started with a Savage Mark II-G, but now my go to .22 is a CZ 457. Very satisfying to shoot.
As for the SKS, recoil is really not a factor. I am pretty sure I am shorter and smaller than you are - I don't find the SKS uncomfortable, not in the least. And I own a few of them and have shot over 4,000 rounds through them. No issues. It is a fun gun to shoot and it may be a decent first gun because you are not using optics and have to use the built in sight and adjust accordingly - again good for first fundamentals development.
For me, the 30.06 is where the ouchies begin...
CCFR, CSSA, NRA (Life), CPC, CHP