Recommendations for Arduino Robotics Kit

It might not be fundable by anyone, but the Lego Robot creator 51515 is awesome. If you want it on the cheap, get this:

At 85€ it is basically Mindstorms (combined with PyBricks.com), buy the RGB sensor with it for 25€ and you have a great starter kit. The brick contains a gyroscope as well, so you can teach Segway like robot line followers and stuff like that with what looks to be a dull Top-Gear car!
This set is similar: LEGO 42124: Off-Road Buggy | Brickset: LEGO set guide and database
The Top-Gear car has better bricks and wheels though.

The great deal with the new technic LEGO is that prototyping is really fast, the new motors and big frames make conceptual building an awesome experience. Starter kits only tackle the brain part, not the construction. Without the 3D printers the school has, things are pretty tough from home.

I would NEVER EVER recommend anyone ANY Arduino Starter kit with LEDs, resistors, the crappy blue LCD's and a bunch of cheap sensors. Especially the Chines clone ones, they are obnoxiuous. Kids get nuts from putting those prototype wires in breadbords for complex electronics, and a LED is a horrible device conceptually with its zero resistance and one-way current direction. They want to start coding right away. Most instructions make things even worse as they ignore pull-up resistors in-chip and start asking to put resistors (impossible with the flimsy legs) or pushbuttons (short too thick legs) in breadboards.

Also great are the Microbit add-ons by Kitronik.

Get a micro:bit, get a buggy and a GameZip64, and you can so some really nice coding: with the Arduino IDE of course, thanks to Sandeep Mistry's great IDE plugin (which is way faster than uploading HEX files from MakeCode!)