For the DC motors, yes, they do need a motor controller, but it doesn’t need to be Adafruit’s. For motors that small, there are plenty of others out there to choose from:
If you wanted to save yourself the money of a motor controller, you could use CR servos. They’re not as fast as the DC motors, but easier to use.
The GVS shield just breaks out the pins from the micro making the connections easier and more secure. You don’t need it, but it’s still better than having loose wires flopping all over the place. You could also substitute any other prototyping shield and solder in your own headers:
There a lot of places that will explain the basics, take the time to check them all out. LMR has some Start Here bots. Some of the better shops like Adafruit/Parallax have tutorials on how to use whatever products/sensors you decide to use. There are robot clubs like DPRG and SRS with an enormous amount of information. And finally, a book I highly recommend you pick up is Gordon McComb’s, Robot Builder’s Bonanza (4th ed). It starts you at the beginning and goes all the way thru a lot more concepts you’ll run into. Definitely worth the money.
You may want to consider consider naming him speedy gonzales or put on the side of him a small sign that says “as fast as fast can be you will never catch me”, boy this one would sure give a cat a run for its money.