You should probably get your feet wet by purchasing a simple Arduino kit with everything you need, and a starter book - and learn the basics of controlling basic stuff, as well as programming the Arduino. Since you are in England, I would look into this kit and the book for it (there's an online PDF for free available):http://www.earthshineelectronics.com/10-arduino-starter-kit.html
Get that - and start playing; learn how to connect and control the parts, and how to program the Arduino. Then you can move on to controlling something that moves.
How you do so is up to you, there are a number of options/ways to go about doing it. You could use a laptop with a wifi connection, and have the laptop control the Arduino (which then controls a motor driver); the laptop might run a web server that you can then browse to using a browser on a smartphone or another computer. Or, you could have an Arduino with a wifi ethernet shield (or just a wifi Arduino), and a small server running on that, which you then control via a browser or something. Or, the Arduino could be connected to a OpenWRT router or something. Lots of options, all different (and there are several more I didn't mention - many ways to skin the wabbit, so to speak).
Finally - before you even go down the route of building such a vehicle - think very, very carefully about security (especially if using wifi) and safety: What will you do to prevent people from "hacking" the communications (if wifi: obviously, you want to set up an encrypted link, WPA2 or something, along with probably adding MAC address filtering and such - but you might want to add a secondary SSL/SSH layer on top of that). What will you do to prevent a "runaway" condition (ie, the motor driver gets stuck, or the microcontroller fails and leaves a pin HIGH) - causing the machine to careen out of control - how do you prevent this? What is the plan if that prevention fails? Where is a "deadman's switch"?
So - when you do get to the point of building your machine, designing the system and software - you should think about this issue of security and safety for the system -first- and design it in up front, before you write a line of code or build anything; in systems design, it is much more difficult to patch in security/safety after the system is built, than it is to design it in up front first. No, it isn't fun; no, it isn't cool - but it may save you some headache down the road.
I should note that this last bit only really matters if the machine travels faster than a jog, or is large - or both; but again, if you see that you might want to increase the speed or size of the system in the future, then you might want to put in the safety/security aspect in up front beforehand (or be prepared to do a complete ground up redesign for the new system).