First time Pure Mages should go for Altmer. 50 extra Magicka, the only Stat boost in the game is an immense advantage learning the ropes. Dual wield Flames, and Altercast pulses to get used to this trick. Watch in amazement how much faster this kills everything except Flame Atronachs (which are immune.)
Run straight for Farengar, or take the cart to Winterhold. You can Save scum for Firebolt, or Flame Atronach from Faralda (or others, pick one of these) then talk to Mirabelle for Novice Robes of Destruction, and Tolfdir for Lesser Ward. (He'll also sell you Oakflesh.) Tool up, and do Under Saarthal before going back to the main quest. Don't forget to grab the staff off the table (In the room with the big Orb, seriously can't miss it.) Wear the Amulet Fragment, and then maybe fast travel back to Riverwood to clean out Bleak Falls Barrow.
When you drop off the stone, buy Lightning Bolt off Farengar, and get out your Ward. This is where timing is a little critical. Wards take a sec to warm up, so as soon as Mirmulnir say "Yol..." put up the ward quick. If you did it right, you will not get staggered, nor take any damage. Hit Highborn for the Magicka to hit him with some Bolts, or just let Irileth, and the boys finish it off. (You can also run the Staff dry pretty quick, if you're out of magicka. The best Enchantment trainer in the game, IMNSHO.)
Hopefully, you healed enough to have some decent restoration, but I doubt you'll be up to Ward Absorb. The important perk is Respite, so you can sprint your Stamina dry, then hit Fast Healing to get it all back. (Really helps with those 7000 steps) Head for Ivarsted, and while in the Rift, you might want to swing by Treva's Watch for Detect Life. (You can buy it, too, but this is an Expensive build, so all the freebies you can get help.) You can also try to clear Halted Stream Camp for Transmute Ore, and make some gold with your Alteration experience. You want Alter/estoration for the perks, Magic Resistance, Respite, Recovery, and so forth.
Here's what i would go for:
Skyrim Perk Calculator - Plan the perks for your Skyrim character before spending them! as a lower level, more efficient build. (to work your way up to the level 50 build already posed.) i wouldn't bother with the perks past Augment Elements, instead, you can add Flames, which is the best damage dealer. Also, either add heavy Armor, or Mage armor, depending if you're planning to wear Gauntlets/Boots/mask, or just depending on your spells. Dual casting is optional, though they can give your armor better duration, increase your healing and lead to Impact, the price you pay is more perks/levels, and less magicka efficiency. Impact is, indeed powerful, but not indespensible for those 2 perks, and much more magicka for slightly more damage.
As for Enchanting, a lot of people are going to tell you, repeatedly, to grind it up, and enchant 4 pieces with 25% Fortify Destruction (You can get another school in there with Extra Effect.) This basically means free destruction, BUT it also means not learning how to play a Pure Mage. it's a crutch, Bethesda put magicka in the game for Ballance, and removing it runs the risk of making it too easy. Not telling you not to, just giving you a heads up. You could also enchant Daedric Armor with this, but it wouldn't be a Pure Mage, like the original question. if you do Enchant, I'd probably focus more on Shock/Magic resistance, but that's me. Magicka Regeneration, and good old fashioned Health are some other good ones.
Fortify health means you don't need to sacrifice points spent on magicka. You do not want to run out of this, because it usually means having your health depleted, and going back to your most recent save soon after. You can get Enchanted Rings to disenchant on the floor of Saarthal, down the hall to the left of Arniel Gane. I can't think of anything better for someone just learning how to play a Pure Mage.
I also like to swing by for the Atronach Stone, where you'll almost certainly get attacked by a Dragon. Give Lydia some good arrows, and she'll help you out, you'll want it's Soul. Once you get to the HHG courtyard, Borri will drop Wuld on you, you can spend the soul on it, and then he'll give you "Nah". At that point, you really shouldn't need carts, nor Fast Travel. Sprint yourself dry, Wuldna foreward, and heal your Stamina. With an unencumbered Altmer, you'll be surprized how fast the miles fly by. If you want to hunt, use Detect Life to find game, then either double bolt it, or hit it with fury, and hit Flame Cloak before it charges in to attack you.
At some point, you should be up to hitting a Dragon Priest, I usually wait for Adept, then hit Vokun in High gate Ruins. The Atronach Stone, High Born, and Ward/Absorb will make his magic fairly inadequate for killing you, it really helps to have a good follower to tank for you. DO NOT SUMMON AN ATRONACH, he Will turn it against you, and try to stay out of Anska's line of fire. Hopefully, you picked up a Scroll of Guardian Circle in your travels to capture, or corner him with.
After that, finish out the DB (for DragonBorn) and CoWH quests at your leisure...