On the space shuttle, engine ignition was delayed for the main engines. Not all of them were started at once, presumably to avoid vibrations. How does that work for the Falcon 9? I imagine that the rather symmetrical layout could cause issues.
Some launchers stagger engine starts for a variety of reasons. As you noted, the Shuttle started the SSMEs one at a time, then the SRBs together.
The Delta 4 Heavy starts one side booster then the middle the the last one, in order to burn off some of the extra hydrogen from chill down. The extra hydrogen has been setting the insulation on fire, leading to awesome shots of the booster on fire as it soars.
SpaceX does hold-down starts, so they run the engines up for 1-2 seconds before releasing, and they seem to start the engines all pretty close to each other; what with 9 starting, any particular sequence would be tricky to get right every time.