So I see this issue raised a lot and never found a good DIY write up. Here is my meager attempt to help the community. If you have a suggestion or a better method post a link.
Why does one need to burp the coolant system? From what I understand air bubbles get trapped in the system causing coolant to not flow some times. This normally presents itself as your car is overheating even though it still might be cold. This happened to me when I installed a new radiator and it took a while for me to figure it out. When I just drained my coolant to refill I did not have this issue. So I think air is introduced when a new part is installed.
So here it goes.
Do your maintenance and then completely fill the radiator. Put the cap on the radiator. Fill the overflow reservoir to the full line. Install a nice funnel into the turbo reservoir. I use a Lisle funnel made specifically for this task.
Lisle 24610 Spill-Free Funnel : Amazon.com : Automotive
Burping my wrx - YouTube
Fill the turbo reservoir until it wont accept any more fluid.
Turn the heater on to its highest setting and turn on the fan as well.
Turn the car on and let it idle. Do not rev the engine. It will make a mess.
Go inside and eat a sandwich or clean up the mess you made in the garage from the maintenance task. Let car idle for about 15 minutes.
Add coolant to the reservoir when you see the level has dropped. Every time the t-stat opens this will let you introduce more fluid. Do this for about another 15 minutes.
If you are daring you can try to remove the funnel now and put the cap on but beware that it might overflow.
The last time I burped my car I had fluid in the funnel that just would not drain down anymore. I simply shut the car off and the fluid drained. I then installed the cap and filled the overflow reservoir a little. Go for a drive and test the car out. You should have nice heat coming from the vents and the temp gauge should be solid as a rock just below center. Keep your eye on the overflow reservoir for the next week. Add fluid to the full line every time you see it is very low. That is it.