All my boys and i did with our 22 Hornet Handi was bench work. Very informal, the most organized we ever got was adding a light-weight bipod, as we tired of resting on jackets, duffle bags etc.
My boys had zero shooting experience before the Hornet and the extra messing about opening and closing forced them to take their time and make every shot count.
Plenty of folks here just target shoot, but I don't how many do it from rests, bipods or off-hand leaning up against a roof post on the range.
The spread between .17 and .223 is big, so most of the folks here will be really helpful and tell you to buy both. I can tell you are on a budget like we were, I would suggest you start off cheap with the .17 and see how you like it, and if you want to graduate to .223, then you can decide then if you are going to sell all the .17 stuff to upgrade or make the .223 a second rifle.
Good luck, and welcome to the best huntin', shootin' and fishin' forum on the net.