It's only messy, if you are not prepared to handle about 5 gallons of fluid. Also a system flush is NOT needed every time, but rather if you notice excessive particulates in the fluid.
1. Wait to engine is cool
2. Open radiator cap
3. open drain (found at open of radiator) and be ready to catch coolant.
4. Once completely drained, close drain.
5. Fill will coolant following direction on container to the full line on the expansion tank.
6. Start car and allow system to "burp" or remove air from the system.
7. refill as needed
8. Replace radiator cap.

its a pretty messy job.. and most likey you will spill radiator fluid all over the ground..

a proper "coolant change" involves flushing the radiator out also... again messy job if you dont have the tools..

advice.. get the correct coolant and the coolant flush from the mazda dealer..

call mechanic shops around and ask how much to change and flush yer radiator if you bring the flush and coolant..