SkyUI is a must. Period, end statement. It gives the inventory system a much needed overhaul.
Before,
After.
Similarly, you'll definitely want to make sure that you have
the four unofficial patches installed to get rid of a lot of bugs.
Moving on to less essential mods,
Realistic Lighting gets a recommendation from me for significantly improving the atmosphere (and actually giving a point to night vision, torches, and magelight).
Skyrim Redone gets one for doing some heavy changes behind the scenes to make the game more challenging, perks more interesting and different playstyles worth trying. Using one handed weapons, for instance, there are perks that make you attack faster when you have a spell in your off-hand, a perk that lets you negate incoming attacks if you're dual wielding and time your own attacks correctly, and one that increases the damage of short sword power attacks if you have a shield equipped, among other things. Or for something a bit more innovative, there's an alchemy perk that lets you make explosives you can apply to your weapons like poisons...or reverse pickpocket onto a foe (which will 1HKO them if they didn't notice you).
If you plan on playing a Khajiit character,
Khajiit Speak could be a nice addition, altering the Player Character's [unspoken] dialogue to be more in line with Khajiit speech patterns and attitudes. The mod page itself has a very good example of this for the start of the Thieves Guild questline when the player is first propositioned. Ordinarily, the player responds "I'm sorry, what?". With the mod installed, this line is changed to "Khajiit is not sure her[footnote]Or "his" in the case of a male character[/footnote] ears heard correctly". The change isn't always that noticeable ("The Whispering Lady? Is she one of the staff, perhaps?" would not be terribly out of place for any race, but is noticeably more Khajiit in its speech pattern than "Who is the Whispering Lady?")