A lot has been covered already...
I suppose my biggest time saver is staying organized. I know where each and every tool I own is at all times. I also have what I call discipline bins, one for painting, one for plumbing, one for electrical, one for tile, etc. I find the higher-quality plastic bins work great for this - I can stack them and they're pretty inexpensive.
One of my favorite tools is my toolbox. It's a Rubbermaid and I can't seem to find them anymore. It's the right size and durable plastic, strong enough to stand on and light enough to lug around (it's still heavy). I put all my general hand tools in it. I'm not a fan of metal or anything fancy - one simple top tray with a deep bottom section is right for me.
In addition to my toolbox I use a job bag. I fill it with the specific tools and parts I need for task so I don't have to lug around my toolbox if I don't want or need to.
I have a drill bag that has my drill, drivers and some bits. I organize that with smaller Klein bags - one for hand-drivers and one for specialty bits and drivers. In this way, I can throw my main bit driver into my main bag and I never have to spend time looking for it because it is not mixed in with a bunch of other small things.
I too love my oscillating tool - I don't leave home without it. We call it the "fine" tool around these parts. Probably because it does fine work, but maybe because Fein makes one too - who knows. Love it for all sorts of stuff, especially making surgical cuts in drywall. Mine is a Rockwell and I've had it for years without replacement.
Full circle back to being organized, the best tip I might give is putting tools back right away and always. I take time to reorganize and clean up a job site multiple times during a day. I suppose some guys are more like a tornado and just clean it all up once at the end of the day or even at the end of a job, but those guys are always looking for stuff - I enjoy laughing at them while they call me slow.
As a tiny-tip, I use a scratch awl to make a small starter hole for screws when doing precision work where it's important the screw goes in an exact spot - like installing a small hinge for instance. Depending on the task, sometimes it works as the starter hole and sometimes it just marks the exact spot for the pre-drill - keeps the screw or pre-drill from getting squirly too. Fantastic when drilling into metal.
Finally I might say the thing that saves me the most time is knowing what I'm doing. I work half as hard and get more done than the young peeps.