Easiest way for my money would be to make use of the fact that the box has three independent video outputs: component, composite (CVBS), and s-video. As nearly as I can tell, these three are completely independent and can be used all at the same time. Just match up with what your second TV has for inputs available. You will want to limit the length of cable, however. The video quality deteriorates with length and long cables actually stress the driving source (the STB).
In order to get audio to the bedroom, you may need to use RCA-type y-adaptors since the box only has one pair of audio outputs.
Being a new-generation box, the 360 does not have RF output which can run into an antenna input on a TV -- those connections are disappearing fast. But if you really do need to run some distance, you can buy an inexpensive RF modulator and then run coax from there to the second TV. Probably not very good quality, but an alternative nevertheless.
Or, you could always ask for a second STB for your birthday....
