There's two main types of cartridges.
The "standard" type is mounted by two screws, set 1/2" inch apart, usually on a removable headpiece. To replace these you would remove the screws and unhook 4 wires from the cartridge itself. You then would replace the cartridge by using two screws and replacing the 4 wires in the proper order, the usual color codes are:
Red = Right Hot
Green = Right Ground
White = Left Hot
Blue (or Black) = Left Ground
Once the cartridge is mounted, you might have to rebalance the arm because not all of these cartridges weigh the same.
Grado is a pretty good cartridge and the prestiege series starts somewhere in the mid thirties and up. Check out www.audioadvisor.com or the needle doctor for these
The other is called a "P" mount. This is a more modern easy to mount type which can be determined by it's single screw, going sideways thru the rear of the cartridge. Once this single screw is removed, the entire cartridge pulls out for removal. You simply slide the new one in and replace the screw and you're in business. All "P" mount cartridges should bete same weight and relative size so it's truly a one size fits all proposition.
Check the needle doctor for these. Likewise, Radio Shack ususlly has some cartridges available at their local stores. ...or at least they used to.