Depends on how big it is? Use a pair of kitchen tongs to handle it. It makes the job real easy. If it's a huge cactus, you'll need another person to help lift it.

If it's a brittle kind, like Prickly Pear, stuff some newspaper gently between the "arms" to keep them from wiggling too much, and then wrap them in an old sheet to hold them together while you handle it. The tag ends of the sheet can be used as a handle to make lifting easier.

Best not to use ordinary potting compost. You can buy small bags of specialised cactus compost which is very free draining. Made sure the new pot has good drainage holes.
(Being picky here - One Cactus, two Cacti...sorry about that lol )