Can you reload 22 shells for cheaper than buying new?

No. The primer doesn't pop out like a centerfire cartridge, so you can't reload. Some accuracy fanatics will hand-load .22 shells, but they buy fresh, primed brass and add powder and bullet.

If you are referring to .22 rimfire, as a general rule, no.

If you are referring to a .223, such as for an AR-15 or varmint rifle, then yes they can be reloaded. I and I know several here reload them, it's a good way to save money or tailor the ammo to the rifle to get the best accuracy.

When .22 rimfire ammo is manufactured, the priming compound used is a slurry. After it is introduced into the casing, the casing is spun to distribute the priming compound evenly. If you have a cartridge casing that has been dented by a firing pin, that spot on the rim in most instances will not be given a hard enough blow by the firing pin to ignite the primer. For reliable ignition, one would have to start with new casings. There are also problems with the crimp used to seat the "heeled" bullets. Not only that, the amount of powder used is so small that it would be easy to make a gross error and overload the cartridge to unacceptable pressures resulting in case failure, etc.
.22 rimfire ammo is cheap enough that trying to reload it would be a waste of time.

I bought a Ruger 22 rifle. Not sure what it's suppose to use. Haven't read the manual yet

I don't want to be rude or insulting but you bought a gun and you don't know what it is chambered in?

Well hopefully it's chambered in 22 Long Rifle and not something obscure like 22 CHeetah Mk II. (that's not a typo either)

I wasn't aware of different ammo for a 22 rifle. It's a standard Ruger 10-22 rifle

Then you'll be wanting to buy some .22 Long Rifle cartridges. Found in every Wally World, Gander and Cabela's in the state. (well almost)

Do not get .22 Shorts, or .22 Longs, or .22 Magnums, or any other variety which does not say ".22 Long Rifle" or ".22 LR". Anything else either isn't safe, or won't cycle/chamber in that 10/22.

A decent price is around $14 for the 550 round boxes, and most CCI's go for around $7 per hundred, but they're considered to shoot a little better than the cheaper stuff.

Unless it's one chambered for .22 Magnum.

One reason I bought it, Salesman said Ammo was cheaper