A few of the mods I run that don't make huge changes to the balance of the game below.
SkyUI - Replace the clunky consolized vanilla UI with a much better one. Also a requirement for a fairly large number of mods due to it's in game mod configuration menu. It is very likely a good idea to install
SKSE first as it enables a lot of functionality and is required by many mods.
Unofficial Skyrim Patch - Pretty much essential. Fixes huge numbers of bugs left in the vanilla game. Pay careful attention to the mod page where it tells you how to set up the load order and follow it. If you have the DLC's grab the corresponding patches as well linked on the main mod page.
This page Lists a number of additional fixes that for one reason or another are not included in the unofficial patch. Several require SKSE, while some, such as the flora respawn fix, cannot be removed from a save that will continue to be played.
Missing Apprentices - Readds a cut minor mages guild quest, almost all of which was left in the game.
Pure Waters - Adds much clearer water textures. There is a version on the workshop
here, but it doesn't support the DLC areas.
Lanterns of Skyrim - All In One - Adds lanterns in the settlements of Skyrim and periodically along the main roads.
Point The Way - Adds signposts at most road junctions so you don't need to keep hopping to your map to see if you are going the right way. May be worth using a texture replacer for the signs such as
Roadsigns Redone as the vanilla signs are not easily readable.
Run For Your Lives &
When Vampires Attack - make civilians flee to safety instead of being suicidal during dragon or vampire attacks. Actual combatants such as guards, companion members, mages, etc will still stand and fight.
Gildergreen Regrown - Makes for a a prettier resolution of one of the Whiterun side-quests.
Bring Out Your Dead - Adds graves for pretty much all named NPC's if they die, in the cities and villages of Skyrim.
Uncle Sheogorath's Really Helpful Hints and Tips. - Replace those dull loading screen messages with far better ones. There was a sequel for the various DLC loading screens but it seems to have vanished off of the net. If anyone is interested in a copy I can put it up on dropbox or something. Also this mod spawned
fanart.
The Paarthurnax Dilemma - Gives you a third option to the conclusion of the Blades quest-line.
Stones of Barenziah Quest Markers - Adds quest markers for what can become one of the more annoying search quests. Toggle it on if you can't recall if you grabbed the stone from a particular dungeon after picking up a dozen of them already, although possibly leave until second or later play-throughs.
One thing to pay attention to if you use SKSE is that you launch Skyrim via skse_loader.exe in the Skyrim directory while Steam is running, not from within Steam itself.
If you use mods from the Workshop when using SKSE, you will need to open the normal Skyrim launcher from Steam in order to download or update them correctly and to set the correct loadorder(unless you set that manually or with another tool, in which case don't click on datafiles).