Can I use larger wire for the run to my shed than what's inside the shed?
The 12 gauge wire in your shed is rated to handle a 20A circuit. This is sufficient for individual handheld power tools but can get overwhelmed if you are running dust collection at the same time or want to install several heavy duty tools (table saw, band saw, lathe).

Running 10 gauge wire means you could support a 30A circuit if you do as @DoxyLover suggested, put in a sub-panel. The 30A circuit would need to run entirely on 10 gauge or better wire. Not patch into the existing 12 gauge wire.

If you don't need 30A now it may still make sense to run 10 gauge wire. Depending on the distance of the run the voltage drop on 12 gauge could be substantial (search on "Voltage Drop Calculator"). Just keep in mind to install no more than a 20A breaker.