DIY 3D-Printed Arduino Robot
This is a nice project that you can continue to work on and modify as you learn more Arduino projects. The Instructable linked below goes through the complete build for the Bluetooth-enabled robot seen in the first half of the video.
As long as you have a 3D printer, this project is really easy to do and the electronic parts cost only around $30-$35. If you try to make the chassis with a 3D printing services, it will cost more than buying your own 3D printer and supplies, so I suggest against it.
It takes in total about 35 hours to 3D print, but can be assembled and programmed in around 30 minutes. It requires only two screwdrivers to build and does not require any soldering. It can be controlled by laptop or Android phone.