For Race: Early on, I find Altmer are the most helpful because you start off with 50 bonus points in Magika. Plus their power is regenerate magika quickly which has saved my ass a few times. Later on, race doesn't really matter though.
Armor: You said you're going heavy? If you are wanting to sneak (which goes well with illusion), the heavy armor might be a pain in the beginning. Have you thought about light armor? It will better allow you to sneak and running away in the beginning (which I do a lot when I start a new mage) will be easier
Or you can use alteration for armor and just wear clothes, but since you are already utilizing two other schools of magic, you will be running out of magika quickly. You said only heavy armor, and you can make that work too.
Enchanting: Get into this quickly if you can, as it will help reduce the cost of your spells greatly.
Conjuration: As someone already said, decide early on if you want to either summon atronaches or dead pl***. If you are a vamp, get the necromage perk and just do dead things. I always prefer my storm thralls once you get to master conjuration because they don't groan constantly (which gets very annoying for me) and are pretty powerful. Quick tip on leveling up: Cast soul trap on a dead thing repeatedly, like a mud crab, if you don't mind leveling tricks like this.
Illusion: Of course you'll start off with fury, try to get calm as well. Leveling magic can sometimes move very sluggishly. So what I do: Get into a room in sneak mode full of enemies. Cast calm on one of them. Then the rest of enemies will start searching for you, cast your atronach or summon whatever (you only get points towards leveling up if the enemies are on your radar), then continue from there. Casting muffle as you travel about is also a good way to level illusion. If you are sneaking, get the perk from the illusion tree that allows all your spells to be silent to others. BEWARE, your illusion spells do not work on undead or dwemer machines until you get the last perk (you need level 90 illusion).
Destruction: I noticed you did not mention destruction. So I assume you're going to rely on followers and conjuration to do your killing for you? (It's actually fun that way). But if you also want to participate in some blood spilling directly, then I would invest in destruction. Try to pick one, maybe two areas, like fire and shock. I normally stick to shock because it helps take care of those pesky mages.
Standing stone blessing: I would either get the Mage stone since it is a pain to level up magic sometimes, or you can get the Lord stone (50points extra armor and resist magic).
Other tips: I would definitely get a follower early on. Once that is not afraid to get up close to enemies while you do your thing from afar. I always like Janessa for the beginning because she does both ranged and close up combat. Always carry around some potions. Get to the College of Winterhold ASAP because you get some very helpful enchanted gear for mages during the quests. Do you have dawnguard? If so, if you join the vamps, you get do a quest for the rings of blood magic or whatever. One of those rings adds 100 magicka and allows it to regenerate quickly, and yes it does work in your normal form, not just VL. As a vampire, you also get 25% bonus to illusion magic and 25% bonus to sneak, if you get the necromage perk, your spells are even more powerful. But keep in mind the sun damage bullpl***. It's a tradeoff, but enchanting can quickly fix that.
Mods: Can't help you here, on xbox!
Have fun!