Complete service manual instructions can be found here. The MR2 has four drain plugs, one in the radiator, two in the piping underneath the car, and one in the engine block. When I drain the coolant, I just remove the drain plug from the radiator, and let it drain. Then I take a shop vac, and put the hose end into the filer neck in the engine compartment where the radiator cap goes. I put the vac on reverse (blow) and blow the rest of the coolant out. This seems to work well at removing the coolant. I then siphon out the coolant from the overflow bottle.
The car must be level for the bleeding to work correctly. My garage floor is not level, but I can use a jack to jack up either the front or rear of the car to make the car level. To bleed the cooling system, there are two bleeder valves that need to be opened. They are both located in the front trunk. One is on the radiator (driver’s side), and the other is on the heater core. After removing the plastic covers, the radiator bleeder can be seen.
Next up is to connect some clear hoses to each of the air bleeder valves. The hose I use for the radiator air drain doesn’t fit quite right, so I use a worm drive clamp on the end of it to make a good seal.
The valves can now be opened. The radiator valve can be opened by hand, and the heater valve can be opened with a large flat screwdriver. Both of them have to be opened about three turns for air to start to bleed through. Coolant is added in the engine compartment filler (where the radiator cap goes), and is added until it starts to show up in the clear hoses up front. Once all the air is removed (which can take some time), install the radiator cap until the first click. It should still feel loose. Start the engine and run it at varying RPMs (from 2000RPM to 3000RPM) with the heater all the way at hot for about 2 to 3 minutes. By varying the RPM, hopefully, the air bubbles will be coaxed into moving to the higher location. I’m not sure if leaving the air vents open during this stage, or closing them results in a better bleed – I’ve tried it both ways with success. Re-open the radiator cap and add more coolant. If you’ve closed the air bleeders, reopen them and check if more air is released. If so, keep doing this until all the air is bled out. The service manual just says to repeat these steps, but it’s a good idea to let the engine coolant cool down some between runs. You may want to close everything up (leave the cap on the first click) and go for a very short drive, to try to jostle the air bubbles to the highest location, and then allow it to cool, and add more coolant. Once all the air is out, fill the overflow bottle, and place the radiator cap on tightly. You may want to fill the overflow slightly higher than the full line as excess coolant will just blow out anyway. This would also be a good time to check the line running from the radiator cap to the overflow, and be sure it is clear/clean.
The overflow bottle can be cleaned without removing it. I just fill it up with a degreaser (simple green / castrol / whatever), and water. Let it soak, and then suck out the contents with a wet/dry vac. Then rinse (or rinse&suck) a good number of times with clean water. I do the same with the intercooler.
The following picture shows coolant in the air bleed hoses while filling.