This is a budget deck aimed at newer players, made as cheap as possible, while limiting it's viability as little as possible. This deck uses 9 basic cards, 20 commons, and 1 rare. Making a budget list for Mage was a bit harder than I originally thought; sure, there's a lot of good basic cards, but stringing them together into a cohesive deck is difficult. I think I made something decent, though. This deck's goal is to get in chip damage through the early and mid-game, and close the game out with burn spells.
You're going to be looking for Mana Wyrm, mostly. Played on turn 1, it's a card that can pull it's own weight, turning your spells into even more damage. Kabal Lackey is a decent keep, but you mostly want it paired with a secret, obviously. If your first draw is bad, then you might want to take a risk. A 1 mana 2/1 isn't the worst, anyway. Beyond that, Frostbolt is a good keep, especially against any match up that has key minions you want to remove ASAP, like Aggro Druid, and other Mages. Arcanologist is also great, although not quite as good as a 1 drop.
Against most classes, your plan is simple. Stick a couple minions on the board, keep their board clear, and get some free chip damage in, using your removal and secrets to keep delaying your opponent. Once your opponent is low enough, you have enough burn, or it's your only option, Fireball your opponent until they die.
Seems simple, but that kinda goes out the window if your opponent has a lot of healing. Priest and Druid, I'm looking at you. These are your hardest match ups, since they are almost always going to be able to recover from any chip damage you deal. Against these guys, you're gonna want to try to hold onto the board for as long as possible, even if you use most of your burn. Your hope is to out tempo your opponent enough that you can finish them in one fell swoop. It's a bad match up, but not unwinnable.
One of the best cards to add to this deck would be Medivh's Valet, if you own Karazhan. Next to that, you would probably want to trade that Ice Barrier for an Ice Block, since it kinda does that "delaying your opponent" thing better. Babbling Book and Blizzard are also decent things to mix in. If you have it sitting around, Bloodmage Thalnos is a strictly better Tainted Zealot. Speaking of legendaries, Archmage Antonidas gives this deck a really good finisher.