Ah, you didn't mention it was one of those types of strip.I'm assuming that you can change the colour of the whole strip with the remote, not the individual segments?The knight-rider strips on youtube are designed that way, with smaller sections sharing a common ground or vcc.Depending on if the strip is controlled as a whole or by segment you need to figure out if they're common anode or cathode.So one live and a ground for each colour, or 3 lives and one ground. This changes the methods best used.If you post up which LED strip it is we can take a look.Have you searched on here for RGB LED strip? I'm sure there will be loads of topics on this already mate
Still not sure if it's common anode or cathode, what connect points do you have?If you cut every 16" you'll be able to control 3 16" segments, can't see you getting a knight-rider out of that mate.I'd cut and solder at least every 2", giving you 24 controllable sections, every 4" if you just wanted 2.
You can tell by the markings on the strip, they'll either say 12v, R, G, B.Or R, G, B, GNDYou can get the whole thing to breathe, flash etc easily enough, you'd only need one transistor per colour for the whole thing