Cheap Shimano Tiagra road bike

Ebay, bikesdirect, etc.

Depends on what you mean by "cheap", but here's another vote for bikesdirect.com. Mine's not even all tiagra and I'm very happy with what I got for the money.

What are the parts that should be tiagra, and what doesn't matter so much?

I have a 2005 K2 Mach 2.0 that I'm thinking about putting up on Ebay. I need to get a box to ship it in first.
It has Tiagra shifters and front dir, with 105 rear dir.
This is the bike.
http://www.rscycle.com/s.nl/it.A/id.8113/.f?sc=19&category=4959

$549 new is a good price. Mine is a size XL and only has about 400 miles, I'll probably get about $300 for it on the Bay.

When you see mixed groups they are usually the lower level except for one or two pieces (typically the RD) so you will see Sora with some Tiagra and Tiagra with some 105. However, typically I would look for the most complicated equipment to be of the highest level so:

1) Rear Derailler
2) Shifters - I would make this #1 if going with a triple
3) Brakes - often overlooked but nice brakes can make a huge difference
4) Hubs
5) BB
6) Cassette
7) Cranks
8) Front Derailler
9) Chain

Basically once you get beyond the hubs, you are looking at weight savings more than anything - the RD and shifters will make the biggest difference (IMO).

Shane