Turning the TV-B-Gone into a universal remote
I don't know much about the TV-B-Gone, but really you're looking to build an modulated infrared emitter. Infrared LEDs are pretty inexpensive, and I imagine they could be directly driven by the PWM available on the Arduino. Range may be an issue, and I don't have the time to do the math right now, but you may end up building a driver circuit where the PWM will drive an amplifier to power several IR LEDs as the per-pin limit is 40 mA (iirc). You may be able to drive the TV-B-Gone this way, but you may very well find it satisfying to build the IR emitter by yourself, and it shouldn't be too hard.

As far as the remote itself goes, my understanding is that the vast majority of consumer electronics all work on a 38 kHz modulation. The protocol used to convey data may vary, but once you get your PWM set to send the data at the right rate that's just an issue of how you turn the LED on an off, which is just a series of pulses that need to be in memory.  One way of capturing this data is, using a working remote control, build an IR receiver circuit (or attach an oscilloscope) to read the signals it sends. This is tedious, but definitely something you can do on your work bench if you try. Another method would be to try and find the breakdown of remote protocols on the Internet. The Linux IR group might be able to help with that, at http://www.lirc.org/. It will require tearing through some source code, and may not give you all the answers you want, but will avoid the tedious nature of brute-forcing the protocols out of existing remotes.

A quick Google query turned up this article on a home-built IR remote control. The article and build itself may not be useful, since it goes a different hardware route, but the PDF attached on the build theory is sound and probably something worth reading (follow the link "build instructions").

Also, the same Google search turned up this Arduino "Universal Remote". A quick perusal only showed it controlling a ceiling fan, but there's lots of good information in this article.

Good luck, and keep us posted on how the build goes! By the time the project is done, perhaps all the information necessary will be in this thread, including a link to common codes.