Well, it depends on what you want and what your macine is capable of (like some old machines can't do hibernation or something; correct me if I'm wrong or misleading in the things below):
- shutdown = halt the system, power down if possible
- log out = log the current user out, showing the (grahpical in this case) login screen, GDM
- hibernate = store an image of what you were doing to RAM, shut the machine down but leave power on (most hardware powered off, though) so that when you next time start up, the image is read and system should be back up in the state where you left it basically faster than a normal reboot
- suspend to disk = same as above, more or less, but the image is stored on disk so the machine can be powered down completely
So, what will you want to do when you "start up"?
- shutdown: starting up is a normal boot, takes it's time, but RAM, caches, temporary files etc. should be cleared too
- logout: just log in; system is already running, so giving your username+password opens your account and you're ready to go
- hibernation: powering on should take less time than normal reboot, and when up, the system should be back at the state where you left it
- suspend-to-disk: same as above, but the image is read from disk; slower I guess. Hibernation and suspend-to-disk don't seem to work on all hardware, or at least the software side isn't working with the hardware correctly, so for example my laptop will happily hibernate but won't restore the image; a startup is just a normal (re)boot.