I used to run little scenarios in Heroscape (similar to HeroClix) for my friends. Heroscape let you build a board out of modular pieces, so I'd do something like build a castle surrounded by hills, and tell my players that they had to rescue someone in the castle. There were no separate story sections, but certain pre-planned events would occur, like guards alerting or a giant showing up.
Mage Wars doesn't have modular boards yet, nor does it have things like trees or rocks. If you wanted to recreate the feel of adventuring through a certain area you'd need a ton of boards and a ton of floor space, so it is probably more feasible to only use the boards for combat as Hey_Daralon suggests.
The first scenario that comes to mind is "A Forcemaster has kidnapped Felalla!" or something, then when you get to the cave where she's hiding out, Fellala has a revealed Mind Control on her (GMs are allowed to bend the rules) and maybe a few other enchantments or creatures around, and the GM plays as the Forcemaster against the players. Throw in a few scripted events (Psylocks swoop from the sides, other defenses the FM set up in advance) and a larger storyline and it could actually be pretty fun.
Since all the players are working together against you I'd say limit their books to 50 points or so, otherwise standard deckbuilding rules. For even more fun, take out the Warlocks as an option when choosing characters, and make the story about uncovering some sort of Arraxian plot.
Personally I wouldn't let the players "level up" their characters' stats at all except maybe once or twice during the entirety of the campaign. Letting them use more bookbuilding points as things go on and they encounter more spells sounds fine, though.