I just bought a XLR to 1/4" TRS cable and as soon as I started using it I noticed that there was a noticeable buzz unless I touched the tip of the cable. If I turn off the external source and mic buttons, the noise goes away. Is the noise supposed to be there or did I get the wrong cable?

1/4 to xlr? XLR to 1/4 I don't understand how u are hooking it up.
It sounds like a grounding issue.

I just hook up my sm58 to the XLR and then the 1/4 to the mic input. I think there is something wrong with the mic so I'll check out my friends to test it out.

You don't want a 1/4" TRS jack - just a mono one. The TRS is balanced like the mic's XLR and plugged into the 404's mono socket will be half disconnected letting noise in (and losing some signal too).

Jim

So the cable I should get is the XLR to 1/4" TS unbalanced right?

Correct.
Although if you or someone can solder you only need to move the wire to the ring contact in the TRS plug to the screen contact . Or if it's easier, solder a short link wire between pins 1 & 3 in the XLR end.

Jim

Hmm I just used my friend's sm57 and the sound is noticeably cleaner. Is the problem still the cable or maybe just my mic? Would the sound be better if I switched to the TS cable?

You should use an unbalanced TS jack whatever mic is used.

If a mic happens to work with the TRS, that either means it's not balanced, has been modified to be unbalanced or a has grounded output transformer inside. Either way, a TS jack will work properly with any mic however it's wired.

Jim

could be a ground loop in your power setup.. anything plugged in in your room/house has the potential to cause a hum... Me, if I get a hum in my mixer, I have to unplug my laptop's power cord.