Revised to make a more standalone answer, rather than rely on the context given by @FalseIdentity's question, and to incorporate some of the commentary.
As with all testing, we want to find problems, so we need to do things what will discover them. But we are also the test pilot, so we have to maintain safety.
Essentially, we need to test every aspect of the bike. Even though a bicycle is quite a simple machine, the list of what and how to test is fairly long.
The first thing to do, as mentioned in the question, is to systematically check that every part seems to be properly assembled and works, while the bike is on the work-stand or still in the workshop. So, as mentioned below, check the head assembly, the front and read brakes, the front and rear gears, the bottom bracket, the wheels. At this stage we are checking that every item is properly assembled and seems to work correctly. As @Criggie says, look at and wiggle everything. I find I need to explicitly carry out this step. Many is the time I've assembled a bike and discovered something later, because I had an interruption at some point.
The next stage is test the bike by riding it. We need to maintain safety, so the first thing is to test basic brake operation. And of course we want to test basic gear operations. We'll be riding back and forth in safe environment, and so we'll be implicitly testing the steering (head set) at this stage.
The first two things to test can be done together, then move on to the others
Start in a low gear, and test the brakes at low speed.
Test the gear changes to a higher hear, go a bit faster and brake harder. Listen, watch, and feel for anything unusual.
As you gain confidence that the brakes are working, build up to testing them with a full emergency stop. You don't need to be going fast, just brake hard. Be sure to test both brakes. Make sure the levers have full travel and don't touch the bars.
Now that you know the brakes work, test all the gears, changing up and down. First without any pedaling force, then under "normal" pedaling pressure. Stop the test if they are not perfect and go back to adjust them. Also see how they feel.
The next test is multi gear changes. Does it change through 2 or more steps at a time. This test depends on the level of gruppo you have. Cheapies have less ability in this area.
If all is well keep testing under more force until you're out of the saddle and changing to higher gears with the rear derailleur. A high end gruppo should do this flawlessly. The rider risk is dropping the chain; this can be painful so take care. A good derailleur should also cope with changing down under such high pressure but this tends to be needed less.
The big gear test is to change both at once. Again this is only for a high end gruppo. Use low pedaling pressure and drop from the big ring to the next smaller at the same time as you change up at the back. Don't do this out of the saddle or with much pedal pressure.
All through these tests you must have been riding someplace safe one way and turning back. Watch for stiff or loose steering.
Check the saddle height. Does it feel right? Be careful in your enthusiasm that you don't set it too high.
Notice how the pedals feel. If they feel loose, lumpy, or grinding then go back immediately to find and cure the problem.
Test the suspension. First with just your own weight, then with small bumps such as kerbs, then with bigger bumps and jumps. Does it feel right? Does it damp properly? Does it recover properly?
If lights or computer are fitted then test them. Computer sensors can be fiddly to get right.
Test the bell :-)
