Discharge all the big caps. Remember, the big glass tube itself is used as a HV cap as well, discharge that too. The inner surface and outer surface are metalized, and the glass is the dielectric. The big fat wire that looks like it is suction cupped is what I believe connects to the inner electrode of this "capacitor." I remember the first CRT I harvested from... woke up about 20 minutes later on my back. Also, there are many wires going from here to there, and lots of exposed metal. Be careful what you touch.
Check your regulators. I've found a few decent sharp 5V regs in stuff that I have hacked apart. Heatsinks are useful too, as they will probably fit your LM317 chips too
Transformer can be useful too, depending on the size.
Flybacks can be fun, I just don't know anything about them.
As far as what's in the tube...
There's this neat assembly in the back, the electron guns. They look like little coiled filaments. I bet they are some kind of special metal! I just kept the thing because it looked cool, made a nice christmas tree ornament.
The screen is glass coated in phosphors, very messy, hard to get off your fingers. In front of the screen is a thin steel foil. It is positioned just behind the phosphored surface, and is perforated. Easily damaged, not much interesting really. Maybe do somethign strange if you shine a laser through it, iono.
Basically, I'd recommend against busting apart the tube, but harvest what you want from the electronics, after taking the correct precautions. Look up part numbers on the internet, you never know when you might find something really nice.
Also, would be nice if you had someone who could check up on you from time to time, make sure you haven't been electrocuted. Have fun!