How do I split a 60-amp subpanel into 3 20-amp circuits?
6 AWG copper wire (x4) for a run less than 75ft., 4 AWG copper wire (x4) for runs less than 150ft.

You can pull individual conductors through conduit, or use underground feeder cable (Type UF).  Individual conductors will have to be rated for wet locations, so you'd likely use THWN.

Since it's a secondary panel, you'll need to run a grounding conductor.  If you're pulling conductors through conduit, you'll want a white grounded (neutral), green or bare grounding, and two colored ungrounded (hot) conductors (black and red are common for hot conductors).

You'll use individual branch circuit breakers in the panel, to divide into smaller (15-20 ampere) branch circuits.

You'll also have to isolate the neutral bar in the panel, so that it's not bonded to ground. Which likely means removing a jumper between the grounding and neutral bars.