I'm a long time player of the 360 version, but recently started playing the PC version, and with mods such as the Unofficial Skyrim patches. I'm new to mods, but couldn't really find any resource that spells out how to properly upgrade mods. My question is what is the best and safest way to update mods? Is it as simple as just copying and pasting the new esp and bsa files over the old ones? I've read stuff about making a clean save, and waiting 30+ days in game, but couldn't understand it.

Any help on the matter?

Totally depends on the mod you are updating. Follow the instructions in the mods readme or from wherever you dled it from. Normally you can just overwrite, but not always. Sometimes you need to deactivate via console or with a clean save. There is no one failsafe way.

A "clean" save is one in which no mods are present or have been used. Sometimes a mod author will refer to a save being clean of their particular mod; in this case, what they mean is either a save that has never had their mod run in it, or one that has had the mod properly disabled and uninstalled.

Skyrim's default reset timer is 30 in-game days. At this point, every cell and worldspace in the game will be reset, creatures respawned, plants grown back, etc... This can be useful for mods like Deadly Dragons to make sure all of the data added by the mod has been reset or removed from your save.

To update most mods, all you'll have to do is replaces the old .bsa and .esp with the new one (this is what the Steam Workshop does, to my knowledge). Every now and then there will be more to the task, but if the mod author is worth their salt, they'll describe exactly what to do in their readme's and on the mod page.