A 23 mm designed rim will most certainly accommodate a 25, but a 28 may cause issues, depending on your bike type. It would be best to ask your shop about that.

Blowouts really shouldn't happen at all unless you hit a blunt object at a high rate of speed. Pinch flats, where the tube is "pinched" as a result of its form being changed as a result of hitting an object. Pinch flats can almost always be avoided by having the correct pressure. You'll see a lot here about lower pressures being more comfortable, and they are, but you need to take your weight and road conditions into account too.

A 28 mm tire can be run at lower pressure, which will make for a more comfortable ride, but again ask your shop. You may only be able to go up to 25 without issues.