Your mage will never be able to handle something like that on her own, instances are meant for group plan and not solo play unless you are a high level running through a low level instance. Actually, when you are on your 55-70 level range you might be able to run through there by yourself. Deadmines is meant for the level 15-22 or so range in a group but most stay out until 18 or so.
Another point I noticed about you when you did not take the minor mana potion, you should take it anyways for two reasons, either use it in an emergency if you are out of all other things or sell it on the AH. Someone always can use things you aren't making use of, and selling stuff will net you money so that you can buy the things you need.
This gets more and more important down the road because your class skills will get expensive, and so your profession skills. Recipies for (Tailoring, Enchanting, Alchemy, Blacksmithing, etc,.) are always for sale on the AH, and they get expensive.
You should go and learn all three secondary professions, talk to a guard and ask the locations for the First Aid, Fishing and Cooking Trainers. Go to the AH and get a greater magic wand, it's a ranged weapon that uses no mana or ammo, a must for mages and priests.
Also go to the Dwarven District of Stormwind and pick up any quests there, and look for the Deeprun Tram, that will take you to the City of Ironforge, a new place for you to visit.
