I don't know how the old editions handled it, but one review of the book mentioned how hermetic mages and shamans used two different sets of rules in 3rd. Well that idea's been thrown out the window - when you make a mage character you have to pick a tradition that determines what sort of services you get from certain spirits (so if you go with the Hermetic Tradition, you'll only be summoning fire spirits for combat services, while a Shaman would summon bear spirits for combat) and how they resist Drain (Hermetics use Willpower+Logic while Shamans use Willpower+Charisma).
As for spells, it works the same regardless of your tradition - pick a spell, pick the Force you want to use for it (basically how strong it'll be, if you pick a Force higher than your Magic stat, you'll take Physical Drain instead of just Stun) and then roll Magic+Spellcasting. How many hits you need is based on the actual spell - some just need a single hit, others are an opposed test against the target's Willpower. Then the Drain hits and if you fail to resist all of the Drain, you'll take stun damage (or physical damage if the force of the spell was higher than your Magic stat, as I said above).
For adepts, most of their powers are just always-on abilities.