What is the best way to make a remote controlled Robot?

Now there are many ways of making a remote controlled bot , I assume u are talking about wireless robots. So here are some honourable mentions... Making a bot controlled over/through Internet using esp8266. U just need a modem around your bot and internet on ur device , so now u can control it from any part of the world. Internet of things. Using Rf modules connected to your arduino. Using raspberry-pi on your bot with a dongle mounted , connected over internet. Gesture controlled bot using Matlab image processing. Gesture controlled bot coupled with physical potentiometers connected to your arduino ,signal being transmitted through rf etc... But the best way will be controlling it by Bluetooth module . It's easy, premade apps are already available on google store such as Bluetooth terminal and arduino rc. Google for hc-05 Bluetooth module. With this u can control ur bot with gesture of ur android phone.

In simple you want a all terrain bot. Few basic requirements would be high torque motors, good suspension, and better grip tires. but the main idea of building a all terrain bot is that it can traverse in any kind of conditions. First thing make sure the chassis is rigid and also light in weight to an extent. Try some suspension mechanism like rocker bogie mechanism, which will suit the terrain and get the other requisites like tires and motors right. You should be able to build one.Hope this helped you, Happy Building.

Well many possibilities have been answered. I would like to add another possibility to it. You can implement Dual Tone Multiple Frequency module for remote controlling a robot. An old technique used in telephones for routing calls. This module is very cheap and easy to implement and has an unlimited range due to the fact that it uses telecom network.

If you do wish to make an all terrain-bot, you need special tyres and an  excellent suspension system for it. Get a chassis, not too big and at a height such that it does not collide with any difference of terrain. You will also need a micocontroller and a motor drive to interface with the motor(Go with Aurdino, I personally would prefer that). If speed is not a factor motor of 100 rpm would be ok. You can use parallel connections for the motor. Then for movement control go with a RF transmitter-receiver system. To power the entire system you will need a battery of 12V. You can assemble it according to your own design such that it looks great or you can look up at some videos for its assembly on internet.

There are many way to make remote control. The communication media cloud be IR or RF. I have made a robot and posted the detail tutorial in my channel.  Hope following videos will help you to make a remote control robot.