What non-Thief skills benefit a Thief character?

This is primarily opinion based, and this answer is based on my play style; which may or may not fit your play style or anyone else.

When I start a thief in Skyrim (which is the primary class that I play) I always play with a Bow. I find it fun to snipe enemies from afar and when you get the cool critical animation from hundreds of yards away, it is something to be satisfied with.

As far as skills and what I focus on first:

Sneaking

This is going to be one of your key skills in order to pull off effective backstabs, and other thieving abilities.

Weapon Skill

If you choose to go with the bow, level up the bow. If you choose to go with any other weapon, make sure you stick with that. Personally, I've tried the two weapon fighting style, and while it's fun, it can be a challenge when facing higher level creatures/people in the game. This is where Enchanting comes in.

Enchanting

This skill is a fun skill to level up, and it's quite easy; as long as you spend the time to do it. I found that Alchemy is a bit easier to level for me as there are more plant and edible things to eat around the world than Soul Gems

I don't really care for illusion spells, and as playing a pure thief, I find myself using less spells than using my bow.

I use the gear from the Thieve's guild as those give me the bonuses I want towards my thieving skills. Whereas The Dark Brotherhood will give me skills in tune with a magic based thief.

The Shadow Warrior Perk is legit. Here is a forum post that goes into greater detail on this skill. Get it, use it, survive.

Don't disregard alchemy and the related skills. Potions of invisibility and fortification are highly useful for stealing and sneaking. Unfortunately, the incredibly overpowered chameleon enchanting from Oblivion is gone, but Fortify Sneak/Fortify Lockpicking/Fortify Pickpocketing are still useful enchantments for your armor.

Blacksmithing is less likely to be useful for pure stealth, but an ebony sword is good to have if you're very bad at stealth!

Here's a list of illusion spells: http://www.uesp.net/wiki/Skyrim:Illusion

Some that seem helpful to me:

• Fear
• Muffle
• Invisibility
• Harmony

So if you're crafty, there are many opportunities that present themselves with different schools of learning.