I can't give specifics because it will depend on the version you play and also it's been a while and I don't remember all mods by heart. So it's just gonna be suggestions; in no particular order:

• First of all you'll need the fundamental bug fixes. There's (still) lots of bugs in vanilla Skyrim.
• You will need the new improved menus, most mods rely on them.
• Personally I can't play without improving the aspect of PC and NPCs, so improvements to bodies, faces and hair are a must for me. If you get down the rabbit hole there's things like mustaches, beards, tattoos, eyes etc.
• Armor and weapons is a close second for good looking stuff.
• You will want a mod that improves polygons as well as something that enhances vegetation, skyboxes, water and weather.
• There are mods that fill the cities and villages with a lot more... stuff. Things like decorative vegetation, benches etc. You will not be able to play without it once you've tried it.
• The skill trees and the professions all need specific mods that apply balances and fixes. You can also go one step further and apply mods that actually make them interesting.
• If you can find one for your version of Skyrim, I strongly recommend a mod that improves dragon AI and makes the fights actually challenging. It always seemed ridiculous to me how easy they are by default.
• Better horses is a good idea, lots of convenience there.
• Smithing improvements. Nuff said.
• Personally I can't stand the default fighting in all aspects of it. I must have didn't roll and some extra brains for the enemies. Some mods the spruce up the dungeons aren't bad either.
• You can get lots of extra quests and NPCs with Interesting NPCs.
• I typically avoid shaders and ENBs in favor of simpler mods that let you adjust the game colors (contrast, saturation etc.) They have very low impact on performance and give you that color jolt that's 90% of why people use ENBs anyway.

On an even more personal note, I like to play like a classic RPG. I get mods that allow multiple companions and interesting NPCs and when I met somebody interesting I take them into my party. There are also mods that let you order them better, you can adjust their flags to set what armor and weapons they prefer, how they level up, and whether they have "plot armor" so they can die for reals. I usually end the game with a party of 4-6 people and it's a blast. But you may want to adjust the difficulty accordingly as you go out you will start rolling everything.

Another very interesting approach I've tried a couple of times is mods that remove all identification clues (no town names, no directions, maximum map fog of war) and start you in some random point of the map. Add some difficulty mods so you have to be really careful who you meet, perhaps some survival mods, and it's a real blast. You can also use rogue rules and restart when you die (and not save scum).