Hi Frog,
This sounds like a normal startup to me. The Compaq logo can sometimes be turned off in the BIOS by entering System Setup when starting up your computer. You would typically see a message on the Compaq screen telling you to press the 'Del' key (or the 'F2' or some other key) to enter System Setup.
In the System Setup (and there are many different versions) you can look for an option to disable the Compaq Logo, and also look for an option to enable a Quick Boot.
To turn off the Windows Logo Screen, you can follow the steps from this article:
http://pcsupport.about.com/od/tipstricks/ht/splash-screen-1.htm
If you have only one user profile on your computer, you can go to Start Button->Control Panel-> User Accounts and select 'Change the way Users log on or off' and uncheck the 'Use the Welcome Screen' option
This should eliminate some of those screens for you and speed up the startup time a little bit.
Cheers!