First, a word of caution - Don't attempt your own oil changes on an aircraft unless you're familiar with the process, and have both the tools and skills you'll need to get it done right.
It's probably a good idea to have your mechanic supervise you on your first oil change, and check your work on the next few until you're sure you've got the hang of it.
This isn't an oil-change how-to - just a list of stuff to look at to judge the general health of an engine.
Disclaimers and warnings out of the way, here's what I look for (or ask the shop to check) during an oil change:
Draining that dirty old oil
First, don't forget to grab a sample for oil analysis while you're draining (I'll talk about that a little later!).
Now, while the oil is running out of the drain you're really looking for big chunks of metal coming out with it (or heaven forbid blocking the drain hole) You also want to make sure there is no nasty sludge coming out (that's a car in the video, but the principle is the much same).
Basically what's coming out of the drain should look like oil. It'll be dirty (dark to black, rather than the pretty clear-ish liquid you poured into the engine all those hours ago), but it should still be flowing nicely.
Obviously chunks of metal here would be bad. If you're noticing them in the drained oil you've got some serious mechanical damage in your engine and you need a mechanic to check it out.
Sludge generally means you need to talk to your mechanic too - often it means a bunch of low-time oil changes to clean out the gunk from the engine, but if it's severe it could indicate other engine problems that need attention
Checking the Oil Screen
If your engine has an oil suction screen that can be removed and cleaned (as on most Lycoming engines), remove it and inspect it for any pieces of metal, then clean it and reinstall it.
As with chunks in the drained oil, bits of metal in the screen are often a sign of a serious mechanical problem - if you find any you're going to want to talk to your mechanic.
Inspecting the Oil Filter
As part of every oil change the filter should be cut open and inspected for fine metallic particles in the pleats. There are a few of ways to do this (one is to simply stretch the pleats out under a bright light and look at them, another is to wash the filter in a solvent and strain the used solvent through a coffee filter to see if any particles are left behind).
EAA has a great video on inspecting an oil filter - it's worth the 10 minutes to watch.
If you find any bits of metal in the filter you already know what I'm going to say -- you're going to want a mechanic to check things out. A very small amount of metal may not be a serious issue, or it might be the beginnings of an engine on a path to self-destruction.
Run an oil analysis
Some people do an oil analysis every oil change, some people don't ever do one.
Smarter people than me say it's probably a good idea to do them, so I consider it to be cheap insurance and a good window into where those "bits of metal" I keep talking about above might eventually start coming from.
There are plenty of companies that do aviation oil analysis and for an idea on what's in the report check out Blackstone Labs' interactive report explanation page.
There's another EAA video that talks about oil analysis which is worth the 3 minutes to watch too...
Note that a single oil analysis won't tell you much (unless things are REALLY out of whack) - it's the long term stability or trends that matter here, so if you're going to do an oil analysis be prepared to do them at every oil change from now on...
Do the things that aren't oil-related
An oil change is a good opportunity to do some other regular maintenance - Since the cowling is off anyway why not clean/gap/rotate your spark plugs (which is another mind-numbing task the FARs allow owners with a private pilot certificate to do)?
In fact since the cowling is apart and your engine is naked now's the time to give it a good once over for any loose parts, chafing wires, etc. that might have come up since the last oil change or service that had someone under the cowling.
It's probably also a good opportunity to check all your lights (interior and exterior), and break out the manufacturer's lubrication chart to see if there are any other things that need to be oiled, greased, cleaned, or sprayed.
So what's all this fuss about metal?
You've probably noticed above that I'm really harping on all the ways you can find metal in your oil in this answer. The simple fact is that all engines wear, and eventually they wear out. That wear shows up as microscopic bits of metal relatively harmlessly suspended in the oil (what the oil analysis picks up), but if that wear suddenly accelerates bigger pieces of metal start getting torn off of the moving parts in your engine, and showing up in other places like the oil filter and screen, letting you know something is wrong with your engine, and hopefully letting you know on the ground rather than in the air.
I don't want to give the wrong impression here though: Not every bit of metal that turns up in an oil filter will be a death sentence for the engine that produced it! In fact brand new engines sometimes produce a few bits of metal during break-in as parts wear in to each other, and they stop after the first oil change or two. The thing to remember is any metal showing up where it shouldn't be during an oil change is definitely something to have checked out by a mechanic.