It sounds like you're viewing "saving the world" and "boring routine missions" as diametric opposites, and you're worried that if you try to avoid the former extreme you'll fall into the latter.
However, these two things are not really opposites at all. It's quite possible to have a boring routine world-saving mission ("Collect the 37 Lost Plot Tokens and return them to the Shrine of World-Saving in order to banish the Sealed Evil in a Can back into the Nasty Place.") and it's also possible to have non-boring missions where the players don't get to save the world (again).
In particular, remember that, even if the players aren't (yet) capable of saving the world, that doesn't mean the world around them isn't going to hell in a handbasket. It need not be obvious yet (because if it is obvious at that stage, then it becomes a survival horror type of game, and I don't think that's what you want in this case), but the world-destroying evil (or several of them!) can be lurking there in the background even in the first session.
Instead of starting your players off on a routine mission of no real consequence (which can work, but doesn't seem to be your style), you can have their very first task already be in some peripheral way linked to the great struggle which it will (hopefully!) eventually be their destiny to bring to an end. Just make their role at that point a minor (but, possibly, crucial) part of the overall scheme, and, to keep things from being too straightforward, make sure they're also beset by difficulties related to their low status, poor circumstances and lack of experience that they'll need to overcome.
For example, you could start your campaign at the point where the players learn that there's a massive high-level conspiracy that threatens to destroy everything they care about, and an equally secretive and widespread counter-conspiracy to which they've just been initiated at a low level. (Or is it really the other way around? How can they tell?) Of course, they won't know all, or even most, of this at first, and what little they do know will always be suspect, but, crucially, they do know that what they're doing will be part of something important — assuming that they can survive all their other problems and actually do it.
That should provide the kind of "sense of global importance" that you seem to want, while avoiding the escalation in power levels that putting the players at the center of the conflict from the beginning would require.
I'm not very familiar with Shadowrun, but from what I know of it, I understand the setting should be full of hooks for this kind of stuff. That said, all I've said above should really work more or less the same way in any system or setting.