Heavy armor. Although I wish I would have started with light, because heavy really slows you down.
You need to find the
Steed Stone; it's northwest of Solitude. Basically required for using Heavy Armor.
I don't undestand why lock picking has to be exclusive against warrior types. And I haven't touched conjuration yet, but I figure I should start having something to supplement 'engage with enemy and hit R1 really, really fast', lol.
Nobody likes passing on loot, so there's nothing wrong with picking every chest you come across. Just don't go out of your way to level it. As you've noticed, the mini-game isn't that hard to master, so there's not much benefit to increasing your skill (and certainly don't waste perks in it); but increasing your skill in Lockpicking has the real and immediate downside of increasing your level, and therefore the inherent difficulty of the game.
As for Conjuration, the lowest level spell that would legitimately help you in battle is
Conjure Flame Atronach. But, as I mentioned before, Conjuration spells are expensive-- even this one has a base cost of
150 Magicka. So assuming you haven't been increasing your Magicka at the expense of Health and Stamina (and for a Warrior, I'd recommend a 2:1 ratio between Health and Stamina) or perking for Conjuration, you won't be able to cast it until your Conjuration skill hits
81. You can cast it at lower skill levels if you've got more Magicka, but gearing for/ choosing to level that will directly detract from your abilities as a Warrior.
I wouldn't recommend it; though if you're determined to nuke stuff, summoning a Flame Atronach beforehand will still allow you the freedom the swing your 2h in battle. So that might be a decent compromise; a couple perk point in Conjuration would allow you to cast it for only 75 Magicka. I've found that Restoration magic is a lot easier to integrate with a Warrior build than the other schools.