Hiya, first off, welcome to the forums!
Now, on to the matter at hand...
Your situation is not an unfamiliar one, returning to the hobby to enjoy it with your children. Be forewarned, it is a slippery slope once you dip your toe in. You have been warned
Based on the pictures you've shared your kit looks pretty mint, are you just looking to get it functional, or are you planning on running it outside? If you intend to make it a runner, you might consider investing in a project car in need of some TLC if you want to go the retro route, or consider buying a new kit. Your current example looks almost too good to run outside and belongs completed and on a display shelf, just my opinion
For new radio systems, what Jeff mentioned is pretty spot on. All modern radio systems are 2.4GHz, crystal-less, and the ones he mentioned have a low cost of entry:
As an example:
https://www.amainhobbies.com/tactic-ttx300-3channel-2.4ghz-slt-radio-system-w-tr325-receiver-tacj0300/p609714
You can go a more expensive route from folks like Futaba, Sanwa/Airtronics, Spektrum, Hitec or KO Propo, but these might have more features than you will ever need or use. Depends on your budget.
https://www.amainhobbies.com/search?s=Radio+System
Interestingly enough, your old servos will work with modern systems and receivers, all you need to do is change out the plug. However, as mentioned, that might not be the best course of action.
Economical servos can be purchased pretty much anywhere.
https://www.rcjuice.com/servos/servos.html?dir=asc&limit=36&order=price
Motors and speed controllers have changed quite a bit, your kit has a mechanical speed control, and brushed motor. Modern systems use brushless motors which are much more reliable and maintenance free. These systems can be had pretty economically as well. Again, depends on your budget.
https://www.amainhobbies.com/hobbywing-xerun-justock-zero-spec-sensored-brushless-esc-motor-combo-17.5t-hwa38020402/p453306
Battery technology has advanced immensely as well. NiMH and LiPo batteries are significantly better than the NiCd batteries we had back in the day. NiMH batteries tend to be more economical and require less maintenance, at the cost of shorter run time. LiPo batteries, tend to cost a bit more, and require certain charging and storage procedures to be followed for maximum life and safety reasons. However, LiPo batteries tend to live longer and will give you the most run time on your vehicle. I prefer LiPo batteries, and the technology has matured enough that a lot of the safety issues have largely gone away if you buy new batteries and follow proper charging and storage procedures.
Hope this helps give you some things to think about. Keep the questions coming!
->Mark