Unofficial patches for the game and any DLC's you have are a good thing to go for since they help to fix bugs and glitches.
Skyui: is considered a must for many but it is only a must if a mod you want has to have MCM menus to make the mod running. I prefer the vanilla UI but there is an option to download a plugin for it that brings back the Vanilla UI and still have MCM. Skyui does need SKSE (Skyrim Script Extender). Quite a few mods do need SKSE.
SKSE:
Since you stated that you want mods that are immersive.
Immersive Hud from Gopher. http://www.nexusmods.com/skyrim/mods/3222/?
Frostfall. Makes the game world feel more like it is in a cold arctic region. http://www.nexusmods.com/skyrim/mods/11163/?
Wet and Cold and Ashes http://www.nexusmods.com/skyrim/mods/27563/? http://www.nexusmods.com/skyrim/mods/31968/?
Those mods make it look like you are in those regions snow piles up on your clothing, You become wet.
Cloaks of Skyrim http://www.nexusmods.com/skyrim/mods/12092/?
I like that mod since it adds a lot of cloaks to the game and since it works nice with Winter is coming if you install this one first and then use the all in one installer for Winter is coming.
Winter is coming http://www.nexusmods.com/skyrim/mods/13486/?%C2%A0
Adds more weather options and also lighting effects.
If looking for overhauls there are quite a few out there. I have only used SkyRe when the last time I played Skyrim. I am starting to get back into the game myself.
SkyRe http://www.nexusmods.com/skyrim/mods/9286/?
That should be a good start for you.
[edit]
Here are some more.
Real Shelter http://www.nexusmods.com/skyrim/mods/52612/?
I have not used this mod but I am looking to use this mod on my next play though. I will need to see if it needs patches for some of my other mods.
Supreme Storms http://www.nexusmods.com/skyrim/mods/27022/?
I have not used this mods but I am looking to use this on my next playthrough.
Splash of Rain http://www.nexusmods.com/skyrim/mods/37873/?
Adds rain effects on puddles.
Footprints http://www.nexusmods.com/skyrim/mods/22745/?
Adds footprints to snow covered areas. A small mod but adds a lot to immersion.