The shopping carts seem to be scattered all throughout the map, but streamers and hardcore fans have found that they can most consistently be found on the race track, in the J-6 region of the map.
They've also been seen in Retail Row and Junk Junction.
To drive the shopping cart, simply walk up to the back of it and press equip, the same way you would if it were a weapon or other item. When you've equipped the cart, a list of controls will pop up on the left hand side of the screen.
The actual buttons used to switch positions, push, and abandon the cart will be different depending on what platform you're playing the game with (ie. Xbox, mobile, PC, etc.).
Like many updates made to Fortnite's ever-changing map, the carts don't significantly change the core gameplay or give any unfair advantage to the players that find them. The basic strategies and skills are still required to do well in the game.
It's too early to tell if the developers have any plans to introduce more vehicles in the game, but the shopping carts are definitely a hit!