Do I get to choose my best mark on the SAT or ACT?

SAT

Yes, partially, and No.
Partially: You can choose your best test dates, not your best section scores. College Board sends scores for all sections from the dates you pick. You cannot pick and choose your best scores on math, reading and writing individually.
Yes: There is an option called Score Choice, which allows you to send universities tests from particular dates.
No: But some schools don’t accept Score Choice, requiring you to send scores from all dates.

ACT

By default, the ACT only sends the scores for the test date you request. You also can request that scores be sent for multiple test dates (for the entire test, not separate sections individually).