There's a handful of empty grand soul gems placed in merchant's inventories from the beginning of the game (for instance, Calindil has one) but the empty ones don't spawn in shop inventories or loot until level 12. The full ones don't spawn until level 15. I haven't checked, but I would assume that the one in Calindil's inventory is physically in his shop (that's generally the case with things that aren't on the leveled lists yet but are consistently in shop inventories), so if you wanted and could get away with it, you could steal it.
There are three of them in a locked display case in the Arch-mage's lobby at the AU. So far as I know, breaking into that case does count as stealing, but it's easy enough to go in at night when nobody's there and take them. IIRC, two are filled and one is empty.
Creatures with grand souls don't start showing up until later levels (generally level 16 at the very minimum), so the only way to fill a grand soul gem before that is to convert it into a black soul gem and fill it with an NPC soul. All filled black soul gems are grand.
Moving back a ways-- a mage with the stock mage class (all seven skills as majors and magic specialization) works out fine. I would suggest dropping one intelligence skill and one willpower skill to minors (alchemy and alteration work well) and replacing them with slow-moving or unused skills, in part so that you've got skills to train for bonuses (since you can get as many minor skill increases as you want without moving any closer to leveling up, so it's handy to have at least one minor skill per attribute in case you need the bonus points but aren't ready to level up yet) and in part because alchemy in particular can increase far too quickly, and thus level you up far too quickly, if it's a major. It's possible to build a mage with only a couple of magic skills (one major per attribute, for instance, or a combination of slow-moving majors and unused majors) and that can work out better in the long run, but it's not really necessary and it's more of a strategy for a patient player planning a long-term character, and this doesn't sound like either one.

Playing a mage is great fun, but it can be a bit of a challenge. You can put him in armor and give him a weapon, but if you want to focus on magic, I'd generally suggest against that. Heavy armor's handy for the first few levels (if it's a minor) since experience in it will give you endurance bonuses, but in order to use it effectively, you have to wear it all the time and you have to spend a fair amount of time getting hit, and pretty quickly your mage could turn into a melee fighter with some magic skills. And no matter how great your skill in the armor is, just wearing it will diminish spell effectiveness, which can be a problem when your spells are pretty weak to begin with (and is also a problem late in the game). It can be rough before you get into the AU, since, unless you happen on some enchanted clothing or are wearing armor, you're going to have 0 AR, but as soon as you get into the AU, you can start enchanting clothing with shield or elemental shield and end up with just as high an AR (or even higher) than whatever you'd get from armor at the same level. So you can either wear armor until you get into the AU, if you can survive using less effective spells, or you can just go for it without armor, but in either case, once you get into the AU, there's absolutely no reason to wear armor (other than roleplay).
So surviving early on requires some strategy and some planning-- you have to learn the strategies of your opponents so that you can time attacks well-- move in, get off a spell, then move back out of range before you get hit, and you have to have a lot of potions, since you will get hit anyway, and when you do, it will hurt a lot. You can use a certain amount of ranged magic, but spells on-target are considerably more expensive in terms of magicka than on-touch, so you get a lot less "bang for the buck," and since you'll be burning through magicka faster, you'll have to have a lot of restore magicka potions. Practice conjuration and illusion to keep enemies at bay-- to make them fight your summons instead of you or to make them fight each other (command or frenzy) or simply to lose track of you (inviso/chameleon). All give you a chance to rest and heal and recharge your magicka or to attack without retaliation at least a time or two.
And yes-- there's a recommendation quest that involves zombies (and another that involves marauders) and both can be quite a challenge for an unarmored mage with generally weak spells. Move slowly, sneak as much as possible so you don't get surprised, draw opponents off one at a time so that you don't get mobbed and carry lots of potions.
Once you get into the AU and can start enchanting clothing and making custom spells, you're well on your way to godliness, and by about the time you hit level 20 or so, nothing's going to be able to hurt you and you're going to be dealing out incredible amounts of damage with just a flick of the wrist.