A rear derailleur designed for 9-speed is no different to adjust than one designed for 8-speed or 7-speed. Sort of like installing a 4x100mm Honda factory alloy on a VW...the process is exactly the same no matter what car the wheel was designed to go on originally.
Shortening a chain requires a bicycle chain tool. Shimano, Park Tool, Pedro's all make good ones, but there are generic ones available also. Nicer ones usually have better handles, fewer sharp edges, more leverage, and are built to a slightly higher tolerance. They are basically a T-handle that threads into a tool which holds the chain. A pin on the end of the T-handle (replaceable pin on nicer tools) drives a single chain pin out of the chain, and is also used to drive the pin back in when re-assembling. If the chain doesn't come with a replacement pin, don't drive the old one all the way out, just most of the way out, otherwise you won't get it back in. Proper reassembly means pushing the pin in until it is centered (and doing so without bending the side plates or links by being ham-fisted.) If the link is too tight, you may need to work the pin back and forth a little bit to loosen it up so that the link doesn't bind, AKA "a tight link."
As far as other parts go, you may want to pick up a replacement derailleur cable in case the old one is crimped or frayed...they should be $3-5. You will also need a 5mm Allen wrench.
As a longtime bike industry guy, I don't spend much time surfing the web for prices, but one of the best resources online for general info and repair guides to bicycle repair and DIY is Sheldon Brown. I think his site is www.sheldonbrown.com but if not, google his name and Harris Cyclery to find the site. He also links to a lot of other sites, so be prepared to spend a few hours surfing! Park Tool also has some good online repair guides.
http://www.parktool.com/repair/
Nashbar, Performance, and REI are all online retailers of bicycle stuff, but the local shop will always give you more information, usually give you better information, and can help make specific recommendations for your bike that you can't get online.
If the chain is broken into mulitple bits, has visibly twisted links, is very rusty, or has links which are bound up and won't move, best to get a new one. A name brand replacement chain should be $18-22 or so. If the chain is intact and each link looks uniform and moves freely, it is most likely fine. If one link is broken and the bike has less than 1500 miles on it since the chain was new, you can most likely put it back together, or put it back together with a new link (about $2) added.
The plastic ring (AKA spoke protector, pie plate, or dork dish) isn't necessary if the limit screw (stop adjustment) on the derailleur is adjusted to keep the chain from shifting into the rear wheel. If you take it off, just be aware that if you crash or the bike falls or gets leaned on the drive side, the derailleur could get bent enough to shift into the spokes, so double-check and re-adjust as needed before you hop on and ride off into wheel damage.