My 2003 Cooper tends to attempt to red line while sitting idle at lights and traffic. Also the heat turns cold until the car moves again. I hear bubbles coming up into the overflow tank but when I try to bleed them out, bubbles or coolant does not come from the bleed valves? Would having the system flush help this problem ?

 

Seems like that would be a good start. We're hoping that low fluid level and air in the system is the problem. Maybe you'll get lucky there if you flush it, refill it, and properly bleed all the air out

Other things you'll probably want to check:
Any leaks?
Water pump working?
Thermostat working?
Cap holding pressure?
Cooling Fan turning on both high and low speed?
Any water/coolant in engine oil or visa-versa?

I'm sure there's plenty of how to any above with a search,.. You'll likely find common problems like plastic tank failing, thermostat housing leaking, and low speed fan resistor failing.

I hope your cooling system problems are not too severe yet.

 

Thank you Alex. Forgot to mention that the water pump was replaced twice. First one manufacturing seal broke. The thermostat housing and thermostat were placed. All new hoses and radiator cap were placed. A new resistor was put in for the cooling fan. But I found out today the cooling fan low speed is not coming on. So I made an appointment to have that replaced. Still going to have the system flushed out. I am also going to replace the expansion tank. Only thing left to replace is the radiator. Thanks again.

 

Sounds like you have a BUBBLE in the cooling system....HOPEFULLY...
Have you tried to idle the car at a slightly higher rpm with the cap off...the tank filled....and keep filling it...and alternate BOTH bleed valves..YES the one next to the radiator and the one next to the thermostat!
you might loose a couple pints or even a HALF gallon of coolant doing this..,, but should do the job...
next...
if that does not do it...
you might be getting exhaust in your coolant from a leaking head gasket...if the bubbles continue....
or if the bubbles stop....but car still overheats....
Some places do use a vacuum bleeder...especially on a NON_S...
you MIGHT have failing water pump PTO gears on the bottom of the SC...usually NOISY.....(ONLY on a S)...

 

I really like my Mishimoto radiator. A little spendy, but their warranty is unbeatable, and it is built solid.