Hi Matt,

Welcome to the forum; plenty of advice to be had on here as you can probably see

Normally I would echo others on here and recommend a featherweight but as you've stated that a) you'd like to have enough power to push a kid around/let a kid ride on it and b) you're not too bothered about a weight class right now, then I think bigger is better in this case.

To have enough power for the 'kid' aspect, either wheelchair motors or scooter motors are the way to go in my view. Wheelchair motors can be quite cheap (like off eBay as you've posted) but are quite slow, so if you want power and speed then the slightly more pricey scooter motors are a good option. In most cases wheelchair motor assemblies are complete, in that they are a motor/gearbox/wheel in one; with the scooter motors, obviously a gear reduction method would be needed. Technobots has a good range of chain and sprockets for a pretty good price and Paul Cooper, the owner, has extensive experience of this sort of stuff and would be able to provide useful information or insights as to what would suit your project best.

What skills/tools do you have at your disposal? If you can weld, a well-designed fabricated box-section chassis would stand up to the punishment that robots take and wheelchair and scooter motors could both be welded or bolted in easily enough.

For radio gear, I'd say you can't go wrong with the Planet 5 system. It's 2.4GHz, and £35 so well suited to a budget. There have been concerns that the signal could be prone to interference due to it being seen as an 'inferior' substitute to the well-respected Spektrum range of equipment, but to date I can gather it has performed well (I currently don't own one myself)
Check out the 'safety issues' thread in the Safety Discussions section to read some posts on the topic.

I'd be best leaving batteries to someone more experienced in heavier machines, as I'm primarily a featherweight builder and am not too sure what is best/required for larger weight classes. You can't go wrong with sealed lead-acid batteries to be honest, but with the low-ish cost of A123s etc, the modern technology could serve you better.
What I will say is that batteries (and speed controllers) is an area where it's usually better to spend a bit more and get a better quality product that will last you longer and can be used in future builds, rather than something that might 'crap out' on you after a little bit of abuse.

Hope that helps you a bit, and good luck with your build