I am new to Skyrim mods. I am downloading the legendary edition as we speak. I have played on the 360 since launch, and have never been able to play a mage for very long because destruction is so weak (imo).

What are the best mods to improve destruction, fix things with an unofficial patch, better bodies, faces, customization, and better clothes for mages?

Rather than post ideas for mods (I'm not a destruction mage), I'll repost some tips on making sure you get the best out of a modded game - if you follow these steps, you'll avoid the most common issues when playing a modded game.

1) Do your research - always read the mod description and readme file, and pay attention to any special instructions for installation, requirements and notes of mod incompatibilities.

2) Use Nexus Mod Manager (http://skyrim.nexusmods.com/mods/modmanager/) to install your mods. Unlike Steam Workshop, it doesn't force-feed mod updates (sometimes you don't want updates, they might add changes you don't want or that conflict with other mods and may break your game) - but it still lets you know when updates are available and gives you the option if you want to update or not.

3) Don't install loads of mods at once. It's tempting, but if things go wrong you won't know what caused it. Install a mod, play for a few hours until you're sure you're happy, then install another...and so on. If I had a dollar for every person I suggested this to but that didn't follow the advice and regretted it later, I'd be a very rich man! Another reason is performance - you need to test thoroughly that your pc can handle everything you throw at it.

4) Never uninstall a mod and continue with the same save game. Your game may break as data from mods gets embedded into saves. If you want to uninstall a mod you need to go back to an earlier save game from before you first installed the mod, or start a new game. This is one of the reasons point 3 is so important.

5) Load order is very important and will avoid issues - download and use BOSS (http://code.google.com/p/better-oblivion-sorting-software/downloads/detail?name=BOSS%20v2.1.1%20Installer.exe&can=2&q=) to make sure your mods load in the correct order.
6) If using mods that edit leveled lists (such as creatures, items, etc), then mods that edit the same lists will conflict and you won't see all intended changes - you need to download Wrye Bash to make a "bashed patch" (unique to your game) which will merge the lists together so you get to see all the new content and make sure all mods work as intended.
7) You'll need SKSE the script extender for some of the more sophisticated mods. This enables Skyrim to do things that it wouldn't be able to do otherwise. You'll be needing it sooner or later, so you may as well grab it as soon as you start - from http://skse.silverlock.org/.
8) Use TES5Edit to clean Skyrim. Bethesda left update.esm and the DLCs all dirty - if you don't clean these then you may get CTDs when you run mods. There's a good video on Youtube by Gopher explaining how to do this and why you need to do this.

Watch Gopher's tutorial videos before you start installing or messing about with mods - http://www.youtube.com/playlist?list=PL20E326325BADAE44 - they're really useful.

You should also use the Skyrim Unofficial Patches.

Other than that - have a browse at the Nexus (http://skyrim.nexusmods.com/), and do a search!

For a more pc-friendly UI, grab SkyUI. It works with either keyboard and mouse, or controller (if you want to use your xbox controller on pc). Categorized favorites is optional - SkyUI is currently implementing its own version of the favorites menu so may be better to wait and check that out first.