What would you start off working on first when starting a new char?

My guy is a Warrior type pretty much....

The first thing I work on when I make a new character is Conjuration, One-handed, Block, Smithing, Restoration and Heavy Armor.

Nothing. I'd just play. Wander around, do quests, see if I can beat the bandits in that cave and if not, then leave and come back later. Join a guild, go hunting, improve my gear.

Melee characters are at a disadvantage until they perk up their armor skill, so take a follower who will help you out. If you're not adamant on being a pure warrior, then think about conjuration - atronachs are extremely useful.

If you're going to fight in the Civil War, joining early might be a good idea. In one of the very first missions you fight against quite a large group of enemies (several groups, in fact) - that's a great moment to level up your combat skills while not putting yourself in danger, because you have your Legion/Stromcloak buddies with you and you can always retreat and let them deal with the enemies.

Do NOT powerlevel smithing, it will gimp you terribly.