Arduino-izing LEGO Technic - ideas for a car / vehicle to choose
I'd be tempted to use the Batmobile (76112) design:



Basically, you have independent motors for the left and right wheels. This allows you to steer quickly and easily without the fuss of a traditional steering system.

The Batmobile is $100 MSRP, and can be found on sale. The included motor controller is Bluetooth based, and can be remote controlled via a smart device or custom software on a laptop. Basically, that set meets your needs if you can afford it.

If that is too expensive for you, I'd still encourage you to consider using a similar design using official LEGO motors. It's relatively complicated to integrate off-the-shelf motors with LEGO (I did this a lot as a kid), and the official motors are actually quite cheap these days. You can pick up Power Functions M Motors for $7.49:



That's $15 for the pair. You'd just need to cut off the Power Functions connector and connect the motor to your driver. This saves you the fuss of mounting your motors and integrating them with LEGO gears and wheels.

This is really outside the scope of your question, but on the electronics side I personally wouldn't use an Arduino. I'd probably go with something that has wireless built in, such as something ESP8266-based. Then, all you need is your microcontroller and motor driver for a working system. The system can be remote controlled from a PC or smart device. You wouldn't even need special software on the remote, as the microcontroller can broadcast an SSID and serve up a web-based controller to the connected device.

So then your BOM would be:


2x PF M Motors ($15)
NodeMCU or other wireless controller ($3)
A few Technic beams, gears, axles, wheels, etc ($5+)
Battery box ($2)
TB6612FNG or similar dual motor controller ($1)


You could probably do the whole thing for ~$30.