What did I do wrong when wiring a new light fixture?
You probably know this already, but the purpose of a switch is to break the "loop" of electricity that feeds the lights.

Find out where all your wires come from and go to, draw a picture of the circuit and that should narrow down your problem or at least help you in describing your problem to someone who could help you. Drawings are very helpful for issues like this. 

From your description, it sounds like the black (hot) wire to the original light was not switched, but the white (neutral) wire was the break in the circuit via the switch. If you wired your can lights up with this in mind, and things still did not work, then there is something else going on that you need to do some investigating to see why your original assumption of where the wires go to/from failed. One of those wires doesn't go where you think it does, or the white/red wire that goes through the switch is not in the same "loop" as the black wire (not likely, nor really possible, but not impossible).

This might be a good opportunity to put the light and the outlets on different circuits. I think that may even be required by code. (ThreePhaseEel suggests that this is only required in bathrooms and kitchens, not necessarily for bedrooms.)

One way to check if you have the same wire (or branch of a circuit) at both ends is to flip the breaker, ground one side of your wire in question to something grounded (pipe, conduit, green ground wire, etc.) and measure with a multimeter that you have 0 Ohms of resistance to ground  at the other end (i.e. place one probe on your wire and the other probe on somethign that's grounded...pipe, conduit, green ground wire, etc.). undo the ground and your resistance should go to infinity (or OVLD or something similar). There are also helpful circuit finding tools that would do this same trick...check amazon.