Disney Infinity

While you can get these the fastest through the Disney Toy Boxes (from the Toy Box, Start -> Online/Multiplayer -> Toy Box Share -> Disney's Tox Boxes), these require you to have a Disney account. If you don't want to sign up, you can get these through the stories.

The car is available in both Monsters and Incredibles.
The helicopter is available in Incredibles, towards the end.
The mount is available in Monsters (the very end).

You'll have to unlock these in order to use them in your own Toy Box, so you'll need to play these missions anyway in order to do that.

The easiest places to get these would be in the prebuilt Disney-made toy boxes.

The Floating Castle has a helicopter for use at a helipad.

The car and horse would be available at the race tracks in the Tron / Sugar Rush mashup.

Both of the previous solutions are outdated, here's how I unlocked this achievement in 2023.

Drive a car - The Blue Autopia Car is unlocked by default in the Toy Box. There's also one already placed down in the Toy Box Launch.

Pilot a helicopter - There are two easily obtainable helicopters available in the Starter Pack, provided you don't have the Calico Helicopter Power Disc. The easiest to obtain is the Attack Copter, which can be unlocked in the Disney Infinity Vault. The second option is The Incredicopter, which is purchasable closer to the end of The Incredibles playset.

Ride a mount - This is the most annoying of the three. The easiest and quickest option is to use a Power Disc, but that requires spending extra money. From what I've seen, the cheapest 1.0 mount Power Discs are Kahn (Mulan), Headless Horseman's Horse (The Adventures of Ichabod and Mr. Toad), and Tantor (Tarzan). All three go for a couple of dollars each, and would also unlock you the Power Up! achievement. If you don't want to fork out the extra dough, Archie The Scare Pig is your next best option. You ride him in the final mission of the Monsters University playset, and afterwards he becomes purchasable.