You will need several things.
First I assume that the power supply is a 12V power supply. If it isn't, better to get one (3.5m of leds can be 4.2A, so even a 12V 5A supply is fine). Note: you didn't specify the led size, so the current can vary a lot. But a standard one is a 60 leds/m 5050 leds, which means 60mA/5cm, or 4.2A for a 3.5m strip.
Anyway you will need a 12V power supply to power both the strips and the arduino (through the Vin pin).
You will then need a way to control the leds from the arduino. Since the required current is too high for the arduino, you will need to put a transistor. I tend to prefer nMOS over NPN, but just because of the way I learned them. Anyway just get a low-side transistor (either an nMOS or an NPN) capable of handling the required amount of current (at least a couple amperes, better if they are >5A) and the proper circuit to turn them on. Usually you will just need one or two resistors. You will need to connect the +12V to the plus terminal of the leds, the drain (or collector) of the transistor to the ground pin of the led you want to control, the source (or emitter) of the transistor to ground and the gate (or base) of the transistor to the arduino through the proper circuit.
Then the remote controller. You did not specify what kind of remote controller you want to use, but usually you can cope with a IR receiver (e.g. the TSOP382 or any other) and the proper decoding. A lot of remote controllers are already mapped in the library IRremote; you just have to read the value it provides; inside the arduino you will need to save the codes for the commands (you will have to train it first) and then simply compare.
This is a generic answer, because you did not provide any details. If, implementing this, you encounter any problem open a specific question with what you did up to that point.