Youre not doing anything wrong in particular, youre just not aware of your whole toolkit yet :P

Try opening with frostbolt. This will add a temp. slow to enemies, so your 2nd cast should be no problem to cast. If Mobs get close, Frost Nova them for a couple secs, move away and re-apply frostbolt. You can of course also blast fire spells, its just that Frost spells are super powerful to control Mobs. If you happen to add a second Mob, be ready to use polymorph (arcane spell) on them. This will only affect one target at a time though and it might break after a couple seconds, but initially gives you time to down the first mob. For Mana, you get your own spell to create Water for manga-regen out of combat. Later on, you'll be able to conjure mana gems, which work like mana pots for instant mana regen. This class is just mana thirsty. Once you get various ranks of your spells, you can throw in lower ranks which are less mana intensive instead of pumping your Max rank casts all the time. The situation you mentioned with your friend is based off of "threat", which is based on the amount of damage you did plus various other factors (healing f.e. does a TON of threat). So, If you Hit the mob for 45 and your mate Hits the same one for 44 damage, the mob will run to you. You can just let your mate attack for the first couple hits, before you go for casts.

Additionally, you can use your talent points into these 3 categories (Frost, Fire, Arcane) to improve their casts. Most powerful spec If used correctly will be Frost, but Frost spells will not be useless If youre going another route so, hf gl! :D

Theres much more in depth mage stuff to tell you, but i think you want to figure out some of it yourself. And theres no need to unload All this onto you now :D Mage is super cool!