Wiping Computer with No Discs

Hi guys, I recently moved to a place with no Internet, cable, or cell phone reception.... Once I got there, I started using my computer for a lot more than just surfing the Internet (movies, messing around with web design....pretty much anything that doesn't require an immediate Internet connection), and I've realized just how slow and bogged down it is!

Right now I'm back in civilization for awhile, so I figured I'd come online and hopefully figure out how to get some stuff on my computer fixed. I found a recommendation to this forum on another forum I frequent, and the user said you were all a great help to him recently.

I have two computers, one laptop, one desktop that I want to get all fixed up.

The laptop is a Sony Vaio VGN-N130G, what I'd like to do is back everything up onto my external hard drive, and format the hard drive. I'm hoping to basically have a fresh start on my laptop. The problem is that I've lost the start up discs sometime during my move. What is the best way to format my hard drive and reinstall everything? I have heard that there is a way to create your own start up discs based on files already on your computer, but the problem is that the disc burner doesn't work. Can I create these start up "discs" on my external hard drive?

The desktop is one that was given to me by a friend. She forgot the password to the admin account, and the guest account is so limited that there isn't much I can do. It is an HP Slimline s7700n, and, of course, she does not have any of the discs. She said if I can figure it all out, I can keep the computer. So, like the laptop, I'd like to just format the hard drive and reinstall everything. I don't need to save any information off of it, as she only used the computer for web surfing.

Wow, re-reading my post, I hope what I want to get done is even possible, thank you all in advance for taking the time even just to read this!

*Edited to Add* I did a search on these forums to see if someone else had had the same problem, but I couldn't find anything with my specific problem.