"So, a couple of you have mentioned that I should replace the chain. I have no problem with that, but what is the reason? Is it because existing chain and cassette have worn down together?"
Yes. See this link:
http://www.sheldonbrown.com/chains.html
"If so, how does that logic apply to not replacing the chain-rings on the crank?"
Teeth on the cassette (especially the teeth on the smaller cogs) experience much more wear than do teeth on the crank chainrings, though it may eventually be necessary to replace the chainrings as well.
Sheldon Brown (R.I.P.) has a nice explanation of cassette removal process; see also the Park Tool website.
http://www.sheldonbrown.com/k7.html#remove
http://www.parktool.com/blog/repair-help/cassette-and-freewheel-removal
Here's a ParkTool link describing how to go about replacing the chain.
http://www.parktool.com/blog/repair-help/chain-installation-derailleur-bikes
Sheldon Brown and ParkTool discuss chain length at the links below. You will almost certainly need our new chain to be two links longer than your old one in order to accommodate a 27-tooth rear cog.
http://sheldonbrown.com/derailer-adjustment.html#chain
http://www.parktool.com/blog/repair-help/chain-installation-derailleur-bikes
You should acquire a chain measurement tool, and replace your chain whenever it indicates you should. Chains are cheap; cassettes and chainrings are expensive. By replacing your chain before it's excessively worn, you can prolong the life of the rest of your drivetrain. As a general rule, any new cassette should start out its "life" with a new chain. Here's a link to the chain gauge I use and prefer:
http://www.amazon.com/ProGold-11022-Progold-Chain-Gauge/dp/B001AYMR7Y
Keep your new chain clean and well lubricated and it will last longer and perform better. Use of a 9-speed SRAM link with your 9-speed chain (whether SRAM or Shimano) rather than Shimano pins makes it easier to remove your chain for a periodic thorough cleaning; in between such cleanings, use a brush (old toothbrush will do) to remove grit and then apply lube. I like White Lightning's Clean Ride self-cleaning wax lube myself, but everyone has their own preference. (I do not wish to rekindle the holy lube wars....)
http://www.whitelightningco.com/products/index.htm
Note for anyone else who may refer to this post: Be aware than SRAM's 8-speed links work with SRAM 8-speed chains only, and do NOT work with Shimano's 8-speed chains, which are slightly wider than SRAM's. If you have an 8-speed drivetrain and wish to use SRAM links, purchase only SRAM chains.