So, about to start my first "real" playthrough of Skyrim (I never consider the first couple of playthrougs "real", they are just learning... )

What mods are considered must-have? I don't care for any quest/story stuff apart from the main game and official DLC at this point, more stuff that adds realism, etc. I have the unofficial patches, obviously, plus Cutting Room Floor, When Vampires Attack and the one that makes citizens run indoors when there's a dragon attack. I'd like something that makes swapping weapons, or switching between weapons/spells easier, although using the 'Favourites' isn't THAT bad, I guess...

SkyUI + better message box controls to make navigating menus less... well... painful. "more hotkeys, please" will do exactly what you described. You can even bind several actions to the same key. Convenient horses makes owning a horse worthwhile. Those are the only non-graphics mods I can think of. For graphics, well, there are several. Skyrim HD 2k textures, Climates of Tamriel, Enhanced Lights and FX, and maybe an ENB Preset if your system is powerful enough. I personally hate the vanilla sound effects so I use the Immersive Sounds compendium to spice up combat and the world in general. Anything else that modifies combat, leveling, or spells I find comes to a matter of taste, so just experiment until you find something you like.

Also, I recommend using Mod Organizer for you mod manager. You will have a lot less of a chance of breaking your install of Skyrim that way.

Mod Manager is very helpful. With that, the S.T.E.P series explains well how to get it set up and working right.

SkyUI, SKSE, FNIS to set up what is needed for many mods to work.

Convenient Horses is very helpful.

For immersion, iHUD, A Matter of Time, and iNeed.

Craftable and Placeable Havok Enabled Bedrolls is great fro letting you sleep when you need to/want to.

I also like Take Notes -- the Journal of the Dragonborn, as a great way to enter into the story for yourself and create a narrative for your character.

Dual Sheath Redux is also nice.

I'm a pretty big fan of Morrowloot and Requiem.

RaceMenu and Solstheim Draugr are nice too.

How is Requiem these days? I don't think it's been updated in awhile, right? I really liked how difficult it made everything, but it seemed like it was falling more and more out of date, so it was getting hard to make it compatible with the rest of my mod list.

There are no such things as "must have mods". As for myself, I wouldn't play without a backpack, Skyrim being "translated" into British English, a mod that changes the placement of quivers and a few others including http://tesalliance.org/forums/index.php?/files/file/1820-whitewater-farm/ . See my profile.

I would use "More Hotkeys Please", it's very usefull for mages if you want them to have an extensive repertoire and avoid going to Menus all the time.
http://www.nexusmods.com/skyrim/mods/23416/?
If you're not so crazy about Hotkeys though, you can use "Categorized Favorite menu" which makes it more like Morrowind, where you could see everything with one key.
http://www.nexusmods.com/skyrim/mods/4862/?
P.S search Categorized favorite in nexus though, cause there are other mods which give you premade profiles, MCM menu and compatibility patches.

Must have:

iHud

Extended UI

No pause between lines

No blur

Not a student

Timing is everything

The choice is yours

Skyrim unbound/Live another life

Race Menu

Thieves guild requirements

Enhanced camera

Highly recommended:

Hunterborn

Frostfall

RND

Cloaks of Skyrim

Real shelter

Expanded Towns and Cities

Can I recommend my own hotkey mod AH Hotkeys?: http://www.nexusmods.com/skyrim/mods/59171/?
By all means, feel free to try them all - sometimes it comes down to personal preference. A few I've used in the past (with varying levels of success) are More Hotkeys Please, Hotkeys Plus Plus and Loadout. There are a few other more specialized hotkey mods as well you may find useful (but I haven't used them).

+1 to this.

A proper STEP install will have your Skyrim running smooth, crash-free and looking good. You can add the STEP packs on to the core install to customise your play experience...