You’ve received excellent information about .22 CF cartridges and their appropriate bullets, but don’t go by word of mouth when it comes to reloading. Go by published data, preferably from more than one source.
It’s best to purchase reloading manuals published by the manufacturers of the bullets you plan on using. Follow their prescribed recipes to the letter and work your way up to maximum loads, watching for signs of excessive pressure. Among other specifics, the manuals will tell you exactly what bullets to use for the various calibers.
FWIW, the 5.56mm NATO cartridge and the .223 Remington cartridge have slightly different dimensions and you can generate excessive pressure shooting 5.56mm cartridges in a gun chambered for .223 Remington. SAAMI lists it as unsafe to do so.