Someone has done exactly this, and written detailed instructions at http://www.instructables.com/id/Autonomous-Control-of-RC-Car-Using-Arduino/
Some remote control devices (usually not the really cheap ones, like the one shown here) will have a standard transmitter/receiver. See http://www.alibaba.com/product-detail/3-channel-radio-control-receiver-box_311659816.html for a picture of the receiver. If you can find this in your remote control device, you should be able to unplug it, and replace it with an Arduino. You will need the "Servo" library - see https://www.arduino.cc/en/reference/servo
You can probably get away without soldering - you will probably need to strip some wires, and you can use crocodile clips, and/or breadboard jumpers to connect everything up. It won't be pretty.
If you want to learn to solder (I know it's intimidating, but it's not so bad) I recommend that you find a local hackspace/makerspace, and find someone there - they will probably be more than happy to teach you. They probably have everything you need, be nice to them, and if you want them to continue, join up & pay regular subs. And take every opportunity to teach the skills you have to anyone else who asks for help.