https://www.youtube.com/watch?v=RGNLyMG9hEk This is a nice project that you can continue to work on and modify as you learn more Arduino projects. The Instructable goes through the complete build for the bluetooth enabled robot 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. It takes in total about 35 hours to 3D print, but can be assembled and programmed in around 30 minutes. Requires only 2 screwdrivers to build and does not require any soldering.
I wanted to design this thing to be inexpensive and use common parts, so this robot uses 9V batteries currently. They run out pretty quickly with a lot of use, so I will probably end up designing a new piece that allows for the use of rechargeable lithium ion batteries and Adafruit power booster.
Also, if anyone knows of a good 3D printing service, so people without printers can make this, it would be much appreciated, I got an initial estimate of over $1000 so I was discouraged from wasting any more time uploading the files elsewhere.