Disney Infinity

Here's an (almost) complete list of the Feats. These must be done in the Toy Box, and many will require you to unlock certain items (like vehicles, mounts, elevators, etc) through the Disney Infinity Vault. You get Vault Spins through various ways, like completing mission events and finding certain capsules.

You can check your Feats from the Start Menu. There are more than 75, so you don't need to complete everything. Not all of these will be visible from the start. They don't always stack, you may have to do something twice (like Dodge Ball and Dodge Ball 2).

The easiest way to:
Mount challenges - go to the Disney pre-made Medieval Arena.
Destroy things - Get a helicopter, place a lot of small spruce trees or enemies in the same place.
Air time - Place a fan and slowly drive your car on top. Also fantastic for leveling characters. (See Grobanite's excellent guide with vSully's fantastic video for the Character Elite Force achievement.)

Building
Creator Extraordinaire - Place 500 blocks total
Game Maker - Place 10 Creativi-Toys in one session
Game Maker Extraordinaire - Set up 100 Creativi-Toy connections
Green Thumb - Place 5 plants
Green Thumb 2 - Place 50 plants
It's Alive! - Cause 10 Creativi-Toys to react simultaneously
Landscapin' - Place 5 Terrain Blocks
Lookin' Good - Place 5 decorations
Lookin' Good 2 - Place 50 decorations overall
Master Builder - Place over 100 blocks in one session
New World - Customize 5 terrain blocks while in Spark Mode or with the Magic Wand
Track Changin' - Customize a track piece (just change the color)

Ground Vehicles
Air Time - Stay in the air for 3 seconds in one try
Air Time 2 - Stay in the air for 6 seconds in one try
Backwards - Get 5 seconds of air time in reverse
Barriers Not Included - Jump over 100 objects overall
Barriers Not Included 2 - Jump over 1000 objects overall
Blast Off! - Ride off a Launch Ramp
Crash Course - Drift into 2 bashables in one try
Crash Course 2 - Drift into 5 bashables in one try
Crazy Driver - Do 2 tricks in one attempt
Crazy Driver 2 - Do 3 tricks in one attempt
Daredevil - Do 4 different tricks in one attempt (all four directions)
Drivin' You Crazy - Do 100 tricks
Drivin' You Crazy 2 - Do 500 tricks
Hop Over - Jump directly over 10 blocks in one attempt
Hot Pursuit - Race around at top speed (boost) for 5 km total
Look Ma, No Road! - 10 minutes in the air
Look Ma, No Road 2 - 30 minutes in the air
Nifty Drifter - Drift over 5 km total
Nifty Drifter 2 - Drift over 50 km total
R.O.F. - Do a trick through the Ring of Fire
Road Rage - Destroy 100 enemies in one session
Sky Driving - Jump over 400 meters in one attempt
Sky Driving 2 - Jump over 800 meters in one attempt
Slide Swipe - Drift into a townsperson
Sunday Driver - Total of 100 km
Sunday Driver 2 - Total of 500 km
Trickin' - Do a trick in reverse
Tricky - Do a trick while going boost speed
Triple Tricker - Do 3 tricks while going boost speed
Turbo Booster - Use 25 boost pads in one session (just put them together on the road)
Turbo Booster 2 - Use 3 boost pads in 3 seconds (drive through them again)
Turbo Charged - Boost 100 times
Turbo Charged 2 - Boost 500 times

Air Vehicles
Air Croquet - Fly through an arch block
Air Croquet 2 - Fly through 5 arch blocks in 10 seconds (just line them up together)
Frequent Flyin' - Total of 25 km
Look Out Below! - Destroy 100 enemies in one session

Mounts (Not just horses )
Broken In - Total of 100 km
Equestrian - Jump over 10 meters in one attempt
Equestrian 2 - Jump over 50 meters with a mount in one attempt
Giddy Up! - Destroy 100 enemies whle mounted in one session
Monster Mount - Ride an oversized mount while shrunk (Use the Floating Castle Toy Box)
Rapid Rider - Go as fast as you can for 30 seconds (does not work for all mounts)
Rapid Rider 2 - Keep going as fast as you can on your mount for 60 seconds
Saddle Sore - Jump over 100 objects total
Saddle Sore 2 - Jump over 500 objects total
Speedy Steed - Ride top speed for 5 km total (does not work for all mounts)

Townsfolk/Enemies
Dodge Ball - Use a giant ball to hit 2 people in under 2 seconds
Dodge Ball 2 - Use a giant ball to hit 3 people in under 2 seconds
Double Dribble - Bounce a giant ball off 2 enemies within 5 seconds (place it over them)
Go Long! - Throw 100 townspeople (you can throw the same one 100 times)
Just for Kicks - Punt a townsperson (X, A, RT, Y) - does not work for all characters
Just for Kicks 2 - Punt 50 townspeople
Pest Control - Destroy 100 enemies
Pest Control 2 - Destroy 1000 enemies
Straight Shooter - Shoot 5 enemies in under 5 seconds
Wrecking Crew - Destroy 25 enemies in one session
Wrecking Machine - Destroy 2 enemies in 10 seconds
Wrecking Machine 2 - Destroy 3 enemies in 10 seconds
Wrecking Machine 3 - Destroy 5 enemies in 10 seconds

Destruction
Blipped! - Destroy 25 Blip Blocks in one session (walk into them)
Bash 'n' Smash - Destroy 500 bashables
Bash 'n' Smash 2 - Destroy 1000 bashables
Destruction Zone - Destroy 5 bashables in under 5 seconds
Destruction Zone 2 - Destroy 20 bashables in under 2 seconds (use a helicopter :-)
Havin' a bash - Destroy 100 bashables in one session
Sharp Shooter - Shoot 1000 objects
Sharp Shooter 2 - Shoot 5000 objects
Straight Shootin' - Shoot 10 objects in one session
Straight Shootin' 2 - Shoot 500 objects in one session

Miscellaneous
Bringin' up Baddies - Use an Enemy Creator
Clown Car - Ride in a shrunk vehicle while oversized (Use the Floating Castle Toy Box)
Floatin' Along - Float for over 30 seconds
Goal! - Knock a giant ball into a soccer goal
Goooooaaal - Score 2 goals in 10 seconds
High Dive - Land in a Splash Pool after traveling 40 meters
High Hoppin' - Jump 12 times in a single Wall Jump
High Score - Score a soccer goal with an air vehicle (getting the toys is luck based)
Hop and Spin - Jump from one Spinner to another (must be a jump, not a fall)
Hover Happy - Travel between 12 fans without touching any other toys (lots of fans)
Next Floor Please - Ride an Elevator (getting the toy is luck based)
Please Stand to the Right - Stay on a Conveyor Belt for over 5 seconds
Prize Collector - 5 Gold Statues in Hall of Heroes (need 5 total characters, level 10 each)
SCOOORE! - Score 3 goals in 10 seconds
Spinning Shot - Shoot an enemy while on a spinner
Supersonic - Use 2 Super Cannons without hitting the ground
Supersonic 2 - Use 5 Super Cannons without hitting the ground (line them up end to end)

Edit: Thanks Zulghinlour for giving me some more!