For learning to shoot, I would go with a bolt action with iron sights. I've taught shooting for a while, and I've come to the conclusion that starting with the basics is best. Semis are fun, and there is no reason not to own a 10-22 down the track.
But to learn with, just a good quality bolt action repeater will do everything you want it to. It will mean you take the time to learn to shoot properly, instead of learning bad habits throwing lead downrange.
I would also recommend a peep sight if your eyes are good, I find them easier to be accurate with than an open V, but to each his own.
You may find a good deal on a used .22 if you look around. If you've got a local range and gun club, they should be able to help you find a reputable dealer.
If you were going to be small game hunting I would recommend a single shot, it makes you be more sure of your shot, so instead of several wound shots, you wait for the kill. I know that will ruffle some feathers, but its what I've experienced, both personally and while teaching.
On a side note. If you have not already gotten your PAL, when you do, take the time to do the extra work to add Restricted firearms to your license. Best to get your hands on as many grandfather-able rights as you can.