Decks, Patios, Porches and Docks - Hints/Tips/Rules for Demoing a Wooden Deck?

20+ yr. old wooden deck - connects house (vinyl siding) to above ground pool. Previous homeowners apparently did NO yearly maintenance and the wood is rotting in numerous places. So we're going to replace it, probably with composit deck. Having never done this before, I just was looking for some guidance about where to start, in what order to dismantle and if there are any specific "dont's" or pitfalls to look out for. They seemed to have used mostly nails in the construction - -however there are also some standard decking screws as well.

Not seeing what you see it may prove difficult. If you want to post a couple of pictures on a site such as photobucket.com and copy/paste the HTML code to your reply post, maybe we can see your problem better.
Generally, go slow. Think through your demo. Naturally you can go ahead and remove the handrails and newel posts if they are not part of the superstructure. Now you are left with decking and steps. Start away from the stairs and begin pulling boards. It will be slow, but necessary to avoid having it all fall at once. I am concerned about the pool. Once the decking is off, remove the stairs, then cut every other joist using a reciprocating saw. Now you are getting close to a collapse, so be careful. We'll wait on the pix to go further.

Rotted structure or appearance?
If the frame (footings, posts, joists) is still structurally sound I would keep it and just replace the decking. You can create an acceptable color match with deck stain, even if you must powerwash the older wood.
The only material that you will find in composite form is the decking. Its not a structural material.