I'm thinking about starting a mage character that relies on Destruction, Conjuration, and Restoration with skill priority listed by the order in which they appear.

Search for the "Gather round Mages..." thread. Best I've ever read on how to be a great Mage.

Already read it, I'm mainly looking for skill increase tips and recommended races.

Maybe you should read it again.

Try spamming friendlies with friendly spells, and hostiles with hostile spells.
Aside from all that, save and run often.

I posted in that thread myself,
When your character is just starting out it is a good idea to carry a staff or two, one with a Destruction Enchantment, and a Staff of Familiar. The staffs are useful for surviving when your Mage runs through their Mana pool, and a summoned creature can be used as a distraction and for setting off traps.
A Mage is the most expensive build type to play because the Spell Books and training cost money. Do not exploit the Enchanting/Alchemy/Smithing, but use them to make useful items that enhance your strengths, and tone down your weakpoints.
You want to give your character a base Magicka pool of at least 200 to 300 if you are not going to use the free spellcasting enchantments.
Lightning spells tend to have more accuracy, but Fire spells are more fun, especially when combined with Runes and Oil slicks. Frost is great for slowing opponents, but most of the opponents in the game have some immunity to the damage.
EDIT using a destruction spell to gather ingedients like Deer Antlers does increase your Detruction Skill, and if your character has an Alteration Shield spell active, and a Summon they conjured, then Alteration and Conjuration will also level too.
Altmer and Bretons make good Mages, but Dunmer can also work.

Just pwn.

Don't be to quick to dismiss illusion,it's probably the most fun school of magic ad it's very useful too,frenzying a big group of bandits never gets boring,and if you're being rushed by a group,any of the main spells could save you. It's probably the only school that dual cast makes sense with early on as well,and the area of effect makes the spells better than poisons or potions. Also it's very easy to level,with calm and muffle,and you get the silent casting perk with it after a bit. You could probably get away with putting 2 or 3 perks into restoration and alteration for a long time,and put 2 into alchemy instead,which can cover all the magic school effects more or less if you have the right ingredients.

I'd drop Restoration for Alteration as alteration has some useful perks for the budding mage i.e. Mage Armor and magic resistance to name but a couple.

Yes, I'm liking Illusion more and more. The only thing is that you really need to commit a lot of perks (like 12) to make it worthwhile, something that kept me away at first. However, after giving it a fair try, I think it's worth the effort and,as you say, it's very easy to level if a little spamming is OK.

i agree ... illusion spells are the best opening in any attacks.. cast fury.. frenzy.. let them fight... if only one cast fear.. let that poor dude run to his other colleagues and they will all return and you can cast frenzy... and destructions spells.. combining with the slow time dragon shout.. as emergency exit so you have time to maneuver..