How robotic does it need to be? If you just want to make an RC car, then by far the cheapest way is to get a 2 channel Receiver/Transmitter set like this, and a motor controller than can plug into it, like this. Note, I am not suggesting these exact ones, I just did a quick search to show you what I'm talking about. But anyway, use some stuff like that, get a servo for the steering and plug it into the receiver too. You of course will need to build the chassis and mechanical parts of the car, but I'm just telling you the electronics cause thats what you asked about.

If you want to make a robot, then you will need to look at PLCs or MCUs, a PLC is sort of a mini computer for controlling machines, and a MCU is a computer on a chip for controlling machines and electronic systems. If you want to make your own RC components, that's fairly advanced, but it can be done, I haven't done it myself, but I suppose you would also need to start with an MCU and learn quite a bit about radio transmission circuitry.

When I do a project of thois complexity, I do it in the order that I can confirm that things are working, that is, start with connection to the computer, then move on to transmitting and receiving. I would start with the electronic components, as they are going to be more difficult than the mechanical. I agree with your approach of using support chips rather than building the circuitry yourself for each function. The next thing to do is to choose an IC for each function you need, read its datasheet and confirm it does what you want, then get one and experiment with it. Before you order everything just do one at a time, for example choose an IC chip that has a diagram you can understand, theres no need to reinvent the wheel here. For example look at page three of this chip datasheet (I am not suggesting this chip, its just an example, a simple google search for RF controll IC will find many like it) If you read the system diagram for the function you want and can understand it you can buy those components which are designed to work together for probably under $10. Once you figure out that part, then you would move on to the motor control, and so on adding one element at a time. For driving DC motors you may want to look at the L298N, or a similar packaged H-bridge. But again, just get one thing working at a time, if you try to do it all in one big shot you'll just end up with a mess.

I should mention that you can buy premade WiFi and Bluetooth modules, some intended specifically for controlling robots, there are a lot of ways you can do this depending on budget and on the quality and capabilities of the final product you need.