Yes, you can extend off the LOAD terminals on your GFCI (Hot+neutral from LOAD, ground from the pigtail splice).
How hard do you expect to use this outlet? If you're using ordinary tools up to about 1900 watts, then 12 AWG wire will be fine. If you expect to max out the 2400W capacity of the circuit on a regular basis, then you are just on the cusp of distance calling for a wire bump, so I would use 10 AWG.
If you want to bury with 24" of cover, you can use direct burial cable such as "UF type". This looks roughly like common NM/Romex, except is gray, and the wires are different and contained differently.
If you want to bury this with 18" of cover above it, you can use PVC conduit. Check the requirements for which segments need to be Schedule 80; particularly where it emerges from the ground and may be subject to physical trauma. Schedule 40 is pretty fragile. They fit together, but Sched 80 has less inside diameter and can hold fewer wires. Three #10s will be fine though.
If you want to bury with 6" of cover above the pipe, you can use Rigid metal conduit (literally RMC; water-pipe-style stuff; I don't just mean any metal conduit that is non-flexible). The stuff is stupid pricey, but on the other hand, you can trench it with a garden trowel. You will need to make a trip or two to the hardware store to have them cut and thread your pipes.