Ok, so I am afraid my hard drive or whole computer will explode. I am getting a new one, so is it possible to just copy my Skyrim saved games on a USB drive, and then just copy them on my new computer's hard drive? Will it work that way? Or is there a reliable way to save them through the cloud in Steam or something?
TL;DR: What is the best way to backup Skyrim save games and put them on my new computer's hard drive if the old one breaks?
Skyrim does use Steam Cloud to back up your saves, however the amount of space provided is hardly enough to store many saves. The best way to preserve your saves is to move the files over to the new PC.
Skyrim puts its saves in the following directory in Windows Vista & Windows 7:
C:\Users\[UserName]\Documents\My Games\Skyrim\Saves
Skyrim puts saves in the following directory in Windows XP:
C:\Documents and Settings[UserName]\My Documents\My Games\Skyrim\Saves
These files are not dependent on anything so you can freely share them across computers, delete, copy, backup, or anything else.
From Where are the save files in Skyrim?
You will probably move the entire My Documents directory to the new computer anyway.
If you have to worry about space then you could also always store them compressed in a .rar- or .zip-file. Depending on how often you have to access your old saves, this is either a very good or a rotten idea.
