I agree with the other comment, the Ruger 1022 is cheap to shoot and is a fun time. I would stay with 22 short or long and not go with the magnum. The magnum shells are quite expensive and serve the same purpose at the 22 long. Any brand of 22 is a good buy, they are durable and last a long time. GOOD LUCK!!!
Have you considered reloading for your centerfire weapons. I found that after my loading equipment was paid for that I could shoot about 3 times as much for the same money that I had been paying for factory ammo.
I have great luck with the Dillon loading equipment.
I have to agree with the others, a 22 conversion kit is the way to go. I used to spend a ton on .223 ammo for a day at the range, now I spend more in gas to get there than I do on ammo.
I'm mostly a hand weapon guy but when I want to just go and plink as you say I borrow my son's Win 22 and for about $20 I go to the county range (free) and shoot all day without the sore muscle or expense of the 306.