Hmm... Ok that puts a different perspective on things. You are going to need a substantial 12V to 5V dc-dc convertor. Not just to power the Nano but the strips as well.
Your strips will need around: 3 strips x15 LEDs x3 colours x20mA per led = 2.7A. Plus a little something for the Nano. So you are looking for a 3A, 12V to 5V convertor.
You are correct about the MOSFETs. But don't buy them without first checking the model with us here in the forum. You are looking for a logic-level, n-channel mosfet. IRL540 or STP16NF06L would be suitable examples, but there are many more available. How many MOSFETs depends on whether you want to control the colour of each strip independently or together. If together, you will need 3 MOSFETs and 3 Arduino pwm pins. For independent control of each strip you would need 9 MOSFETs and 9 Arduino pwm pins. Uno/Nano only has 6 pwm pins. But it is possible to simulate more pwm pins in software, as long as the Arduino is not too busy.