I would avoid using a pressure washer because of the damage it could cause to the wood. Chemicals such as TSP and/or bleach work well, but these are harsh and allegedly they can damage wood fibers.

There are many brands of deck cleaners to choose from. I've used DeckScapesî Deck Wash in the past, but I don't remember how well it worked.

DeckScapesî Deck Wash
http://www.sherwin-williams.com/homeowners/products/catalog/deckscapes_deck_wash/

For a stain, I would go with a quality stain product from a REAL paint store (e.g. Benjamin Moore, Sherwin Williams).

Here is a link that might be useful: Best Deck Cleaners

If you use a contractor-grade cleaner or put 1 gallon 12% bleach in a 5 gallon bucket and add 2 cups of Dawn and the rest with water....you can spray it on and let it dwell (stay on the wood) for 5-20 minutes depending on the amount of mold/mildew, brush it a little and give it a good rinse down...this should take care of you. You will damage the fibers in the wood using too much pressure so the key is to use the right chemical treatments to do the work. Now...if you are really green with the environment....trying to use green chems to do the work will take longer. Remember that the bleach you get from the grocery store is probably 4-6%....go get some bleach from a pool store. Make sure you use safety precautions when working with strong bleach! Goggles...gloves.

You can also look for some DIY cleaning kits on Amazon or Ebay because you won't find contractor-grade cleaners in the big box stores. And I agree...a REAL paint store (as mentioned previously) may be able to provide some assistance or direct you to the best products.