Irfanview is an absolute must if you do any image editing, as it can handle virtually any image, video, or audio file type under the sun. Make sure you download both the program and the plugins, as the plugins massively increase the versatility of the program. It's also completely freeeeeee.
NCH's Wavepad is a fantastic program if you want to do any sound editing. The free trial only has the occasional nag screen to make you upgrade to the paid-for version.
NCH's Switch program is also great if you want to convert a sound file into another file type.
Audacity is another great sound program that also lets you capture sound direct from your own speakers. It's completely free and can save to a number of formats. Make sure you follow the online instructions to download and install the Lame codec if you want to save files as .mp3 though. (Lame is the name of the codec, i'm not insulting it lol)
Winzip is incredibly useful if you want to package up files in an archive. The website tries to force you into buying the program, but you can download an evaluation copy if you look hard enough on there. And they have a trialpay system as well where if you sign up with an offer for another website, you can get the program in full for free. Winzip has a very easy-to-use interface though so i'd definitely at least recommend the trial.
WinRAR is the main alternative to Winzip. It's free, although the interface is horrible for people that aren't used to it. Many people claim that WinRAR is actually more powerful than Winzip. WinRAR can certainly handle more filetypes, but in reality I always find that Winzip is the better program.
Winmerge is a handy program if you're modding and comparing a lot of text files. It displays two text files side-by-side, then highlights any differences between them. It's free, and is really useful when you're trying to track down typos and bugs.
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