I get a small tub with a mixture of bleach and water, mostly water of course, whatever ratio it says on the bottle. Get a fan and open any windows so it's well-ventilated. Clear out everything you can from the bathroom. New sponge, not a scrubby pad because you'll tear up caulk and paint scrubbing too abrasively. Then I take off my clothes because I'm not about to get them bleach-stained (no this has not just taken a turn for the sexy, you could just wear clothes you don't care about ruining) and scrub the mildew. Get all the surfaces, including stuff that doesn't look mildewed. Start out with rinsing everything with bleachwater from the tub and then go back and go over it again to scrub with the sponge. Make sure you do the floor! Check any wood in the area for mildew, such as the part of the door frame that the door touches when it's closed.
If you keep up with it, the mildew won't cause any lasting damage. But if you let it get too bad, it will get in under the paint and you'll have stains forever. Until you repaint I guess. It can also get under caulk if it gets really bad.
Remember: lots of ventilation.
I also use those yellow rubber gloves for washing dishes, because otherwise my hands get light chemical burns from the bleach. I also wear safety goggles so when I wash the ceiling I don't get bleach dripping in my eyes. I haven't had a single problem yet with chemical burns on my skin otherwise, and nothing has blached my hair. If you care about something like that, get a chemical suit and shower cap type of thing. But at that point, get a respirator and triple the bleach ratio to really kill off the mildew.
Next, maintenance. Clean up any mildew that appears RIGHT AWAY. If the bathroom has a window, it has to be open when you shower. If there is a fan vent, it has to be on when you shower. Preferable to have the bathroom door open and other windows in your place also open to vent moisture from a shower. Some bathrooms just have moisture problems unless you really stay on top of things.
If you want to go all hippie on it and not use any stiff chemicals, the vinegar and baking soda thing does something but I'm not convinced it works any better than water and elbow grease.