First thing I do is download TESVEdit - http://skyrim.nexusmods.com/mods/25859/?
This lets you clean your mods/official DLCs to remove dirty edits that might cause issues down the road.
Second thing I do is download BOSS - http://code.google.com/p/better-oblivion-sorting-software/downloads/list%C2%A0%C2%A0 is the actual download for the utility, more information about BOSS is on its Nexus page at: http://skyrim.nexusmods.com/mods/6/?
Basically it sorts your mods properly so there is the least amount of conflicts/overwriting.
I also highly recommend using Nexus Mod Manager (NMM) for downloading/installing mods.
Now, regarding the Mods themselves - Start with the Unofficial Patches. There is one for the main game (USKP), and one for each of the DLCs, including the High Res Texture Pack. The USKP is at: http://skyrim.nexusmods.com/mods/19/?%C2%A0%C2%A0%C2%A0
It makes the forests feel ancient and epic. Using NMM, there are some customization options.
Another must-have for me is No NPC Greetings - http://skyrim.nexusmods.com/mods/746/?
It affects how close the NPCs have to be to you before they start spouting their life story.
Three files; pick one. One removes the greetings completely, the other two change the distance.
For sound, I like "Sounds of Skyrim" - http://skyrim.nexusmods.com/mods/10886/?
Adds a lot to the atmosphere. The link above is specifically for "The Wilds", if you click on the author's name it will bring up his other mods; there are Sound mods for Dungeons, Cities, etc.
General good mod - Even Better Quest Objectives - http://skyrim.nexusmods.com/mods/32695/?
Expands the quest objectives in the journal so you can play without quest markers if you like, and just improves the detail overall.
Overall Graphic Improvements: SMIM (Static Mesh Improvement Mod): http://skyrim.nexusmods.com/mods/8655/?
And, if you don't like the UI, there's SkyUI - http://skyrim.nexusmods.com/mods/3863/?
**This requires SKSE (Skyrim Script Extender), available at http://skse.silverlock.org/
If you DO like the UI (or don't like SkyUI) BUT you want to use the MCM (Mod Configuration Menu), there's
SkyUI-Away - http://skyrim.nexusm...m/mods/29440/??/a>
Basically you install it after SkyUI and it reverts the menus back to Vanilla while keeping some of the additional functionality of SkyUI like the MCM.