It's great to see that you're asking the right questions
You're right on track with the economy of the .22 ammo. Ammo of the .22 LR type (LR = "Long Rifle", though it is used for both rifle and pistols) runs about 2 cents a shot. 9mm averages 10c/shot, and .45 ACP (Automatic Colt Pistol) runs 20c/shot. So economy definitely favors the .22 LR.
So far as to which pistol to get, three options:
1) Semi-Auto: Ruger MkII and Ruger 22/45 are widely-respected classics, usually cost $200-$300, depending on which model. Great triggers, ludicrously accurate, very good buy. The adjustable-sight models are definitely worth the small markup, and the stainless steel models are very easy to maintain.
2) Double-Action Revolver: a DA revolver is one where you can cock the hammer (to get a light trigger pull) and then pull the trigger, or simply pull the trigger (simultaneously cocking and firing, which takes more effort). These are the kind of revolvers cops used before going to Glocks and the like. Good quality DA revolvers tend to be a bit pricier than SA or Semi-Auto. The S&W 617 is the main example of this family in .22LR. Ruger and Taurus make less-expensive competitors. If you buy a used revolver, check out the Revolver Checkout procedures, the first post in the "Revolver" forum of this website.
3) Single Action Revolvers: These are revolvers of the "cowboy" type. Very simple mechanism, very durable. There's no "simultanous cock/fire" option, you cock the hammer for each shot. The hands-down favorite option in this category is the Ruger Single Six (around $275 and up). Kind of retro, but lots of character and extremely well-built.
I would shop around, see which of the above types suits your taste. If at all possible, find a range that rents firearms, and give a test drive on the type that interests you.
Make sure you know what you're buying! You can always come to this site, hit the "search" button at the top of the screen, and run a search on whatever brand name or model interests you.
Things to avoid: brands like Lorcin, Phoenix, Raven, Jennings, Bryco (cheap, pot-metal guns). Avoid guns with really short barrels (harder to shoot accurately).
Things to look for: Good quality brand names (a firearm can last for a century, the extra $50 in price isn't a huge deal in the big picture). Adjustable sights help a lot with .22, as different brands of ammo will shoot differently. Adjustable sights can be recognized by little adjustable screws on the rear sight.
Good luck with your decision. Feel free to post questions and run searches of our archives. If you have any questions re: the Single Six or Ruger semi-autos, I've owned both, so feel free to message me by clicking the "PM" button below this text.