If you're just going a short distance, you could just hold the robot on your lap in the car without a box. This isn't ideal, as robots can be sharp and heavy, and nobody wants to hold a robot all the way home. However, this is much safer for the bot than setting it down in the car without a box. If the robot is simply in a car without any protection, the jostling during the ride can easily destroy axels and break mechanisms, making for a bad competition day. The more bumps hit, the worse.
If you choose to use a box, there are several options. You could build your own, buy a robox, or buy a robosource robot box. Each has positives and negatives. Roboxes are typically soft-shelled, as shown here, which allows them to be unzipped. They also come with many pockets, which are helpful in storing batteries.
Robosource boxes are hard shelled, which makes them much more resistant to travelling long distances. They have foam padding on the inside, which is helpful for cushioning the robot, but can easily be torn after a few years of use. We've found these are ideal for shipping robots to worlds.
Though we wouldn't necessarily recommend it, you can also build your own hard-shelled boxes. These can save cost-wise, but can take a lot of time and effort to put together. We originally made one when we first qualified for worlds years ago, but it's incredibly heavy. We ended up spending a bunch on foam siding, anyways.
Adding wheels makes it so that much smaller team members can transport the robots, which is a lifesaver at competitions. It also speeds up transportation.
When adding wheels, consider adding them further to the outside of the robot rather than clustered on the inside for added stability, as shown.
Another option is using specialty skateboarding wheels to add stability and absorb shock. These wheels are called Shark Wheels, and they are designed to easily roll over cracks and pebbles, which is perfect to absorb shocks that can effect the robot!