I have a car that has been parked in a shady damp spot for a while and has developed mildew on the exterior. What's the best way to get this junk off?

Keep car in shady spot, work on one section of the car at a time, eg, bonnet & front, left side, roof, etc (to stop chemical drying and marking paint. This creates lots of work). Spray one section of dry car (don't hose car down first, it dilutes chemical too much), with mild degreaser. Even tho it usually says not to spray on paint, you can, just don't let it dry. Leave for up to a minute, and then start washing with a sheepskin mitt and auto wash. Then rinse the section. Sheepskin won't scrath the paint, and is xlent for chrome too. Don't use laundry powder as soap, it corrodes. Rinse whole car again, chamois dry, then polish. Ask your local autoshop what good polish they recomend. I like meguiars.
Good Luck.

Thanks! Do you have a recommendation for a degreaser as well? I'm a bit new to car care. In the past I usually just had someone else do it, you know? Thanks again!

I suppose meguiars would have a degreaser, but I don't use it. I don't know what country u are in, but in Australia, I use a brand called Batch Chemicals (based in Victoria). Any well known brand should be okay, not necessarily the most expensive one. You could talk to a couple of Detailers or Panel Shops, ask them what they use. Its generally cheaper to buy about 20 litres of whatever you get, then dilute it down in a spray bottle, about 1:3 (1 chemical, 3 part water). You could also use it neat (undiluted) on wheels, but make sure it doesn't dry, or your new hobby is going to be polishing the tarnish off. Oh, also, don't spray degreaser (neat or diluted) on to aluminium, like bullbars and stuff, because it also tarnishes.
I'm not sure how to prevent it from happening, apart from not parking your car in damp shady conditions. But washing it shouldn't be too hard.
I just saw your other post in datailing, re the seals and rubbers. There are rubber seal conditioners that you can buy, to help keep the rubber soft. It comes in a bottle like those shoe polishes with the little foam pad thing on the top where the colour/polish comes out. I haven't used them on old rubber, only new, but I'm told they work well on old stuff. They don't stop mildew though, just keeps them soft.
Don't be afraid to go out and ask questions at shops and panel shops, etc. They may be male dominant areas, but most realise how much business us Chicks can bring them, and so are generally really helpful. On that note though, it never hurts to get a second or third) opinion about anything, even product. Beleive me, I've worked in the Auto trade for quite a while now.
What sort of car have you got?