I don't know about the HP (is it vista?), but my Lenovo S10 with XP Home starts up reasonably fast.
In either case, you can use hibernate instead of shutdown to start up even faster. In XP, you hold down the shift key and the first button on the power-off screen (Stand by) changes to Hibernate.
Hibernate saves all state to disk, so it doesn't use any power when it's hibernating. Next time you turn it on, it just loads the saved state from disk.
If it has Vista, Vista's startup is supposed to be faster than XP. It defers loading a bunch of stuff until it's up, so it will be a little busy while it keeps loading the rest of the OS, but it's supposed to be usable at that point. You can experiment with it to see if hibernate is better than a normal startup on Vista. I suspect it probably is, but I've never tried Vista.
-David