All solutions are imperfect, obviously, especially in the days of cell phone cameras, and it is basically assumed that you will lose your secret identity eventually, either to being unmasked by supervillains, or to fans successfully speculating online, or to some jerk with a cell phone. The key to delaying it as long as possible is finding somewhere to change you won't be noticed and nobody will wonder why you're going in, or care enough to track you going out. Bathrooms and changing rooms are useful, especially if they're behind a corner so no one can see who comes in; here's a guide on spotting cameras, it's good to pick places where there are windows so that you can get out a different way than you come in. Your house is best as long as you have a way to depart from it quickly without alerting all your neighbors, and heroes with good travel powers often stay in a lot; heroes with good travel powers and DIY skills will often find some way to get from their house to somewhere a tenth of a mile away without anyone noticing, and there's discussion of some of the DIY skills.
And the discussion of getting around if you can't fly is pretty extensive, but unfortunately is a lot of 'yup, it sucks'. The basic problem is that supervillains mostly don't want to get into fights with SWAT teams (the ability to shrug off bullets from a handgun does not necessarily protect you from tinker-made stunners, anti-materiel rifles, knockout gas, or any of the other things SWAT teams have started carrying since supervillains showed up), so they try to time their heists so they'll be gone before SWAT arrives. And SWAT tries to arrive as fast is as possible without superpowers. The obvious conclusion is obvious, but also stated: It's really hard to be a superhero without travel powers. The ones who make it mostly either have someone with travel powers help them go faster, really good information sources (usually either a product of superpowers or growing up somewhere that gives connections with a high-crime community) so they can arrive before the police learn anything's happened, or specifically show up to the kind of heavy-duty supervillain who anti-materiel rifles really don't help against. This last group (especially if there's multiple of them in a city) will often end up specifically based in an HQ somewhere central with its own flashing-lights emergency vehicle, hanging out while they wait for crises.