What I have covered thus far is cleaning carpet, linoleum (or other moisture resistant surfaces), and furniture.
A good example of a mess-making pet:
Step 1: Getting the Pet Out of the Way
Depending on the surface, you might want to skip to a certain step.
Step 2 is on Cleaning Carpet
Step 3 is on Cleaning Linoleum
Step 4 is on Cleaning Furniture
Well, at least the dog is outside:
Step 2: Cleaning Carpet
-Paper towels (cloth will work, but paper is advised)
-Some kind of carpet cleaner
First thing is to take care of any solid chunks. Grab a towel of some kind (again, I advise using paper so that you can just throw it away when this is all over), and wipe up what you can and throw it right into the trash. If it was urine, grab another towel and fold it so that it just barely covers the spot so that it can absorb more moisture. Apply lots of pressure until most of the spot is gone, then spray the carpet cleaner and let it sit for about five minutes. Grab a new towel, apply more pressure to the spot and then let it finish air drying.
Although this worked fine for a darker carpet, it could be that a stain is noticable on a lighter one. First try to spray it again, maybe with a different cleaner, and if that fails then it might be necessary to rent a carpet cleaner, assuming you don't own one.
A paper towel folded too small (imagine the darker area as wet carpet, we haven't had any real problems in a long time):
Step 3: Cleaning Linoleum and Similar Surfaces
-Spray cleaner (like 409)
If your pet made a mess on linoleum, It should be fairly easy to clean up. After following the first step, just wipe up any solids with a paper towel, and throw it all strait into the garbage. Then spray the area down, and wipe it dry. That was easy!
If you feel the need to get it really clean, you could mop the area down, but it's probably not neccesary.
Step 4: Cleaning Furniture
This one could be the hardest to clean. Again, remove any solid chunks and try to soak up any moisture. Depending on the fabric, construction, etc. this might not work very well. Spray it down thouroughly, and try to let it sit long enough that it can do some good. Again, depending on the type of fabric and cushions it may or may not work well. Put the parts that can easily go through the wash (covers, etc.) in there. Beyond that a healthy dousing in air freshener can't hurt.
A bad sofa to get messed up (the cushions don't come off):
Step 5: General Tips and Tricks
When using a spray bottle, set the bottle to stream, and spray from a distance. Why? because as the bottle gets towards half full, the straw part is no longer submerged in liquid, and won't spray (the first image shows how I would NOT advise spraying, the second one shows how I usually do it
Besides that, don't forget to empty the trash, as it could get very stinky!
As pointed out by Erik, a one to ten mixture of diluted bleach water can help get rid of stains.
This is the biggest area for people to add their ideas. Also I'd like to hear any comments/criticisms that people have.