Let engine warm up before driving it hard. You can drive it immediately after startup as long as you drive it gently, which means half throttle or less and 3000rpm or less until it warms up (about 5 minutes).
Never start the engine cold and shut it off before it warms up (for instance to back it out of the garage etc.). Let it warm up fully before shutting it off.
Redline the engine at least once every time you drive it if possible (after being fully warmed up, preferably shortly before you shut it off). This will serve two purposes. 1) it slings carbon off of the rotor faces and 2) it exercises all the intake valves that rotate at certain rpm levels, keeping them from getting stuck.
Avoid short trips if possible, the engine never fully warms up and enters lean fuel burning mode, so this can lead to increased deposits in the engine and shortened life.
Excessive idle time and low rpm driving (especially in automatic models) is bad for this engine. It leads to carbon deposits in the engine and intake manifold which reduce performance and compression and kills the engine. This engine is most efficient in the 3000-4500rpm range and that is where you should spend most of your driving time.
Oil is very important to this engine. It does up to 40% of the cooling as well as the lubrication of both the conventional bearings/shaft and the auxiliary oil injection for the apex seals/housing compression surfaces. Oil breaks down quickly in this engine and this is why we cannot stretch oil change intervals out as you might do in another kind of car. Check and change the oil very regularly. Also check that the engine oil coolers, mounted under the headlight(s) are not clogged with leaves or debris, and consider straightening the small fins between the cooling rows to restore proper airflow through the coolers.
rotary engines are highly intolerant of overheating. You get one overheat with this engine and then it is ruined from then on. IT is not like an old pickup that you can overheat, drive home, put water in it, and have it be undamaged. IF at any time you are losing coolant or suspect the engine is getting hot you must shut it down NOW and find out the cause.
Change engine oil every 3000 miles or less (if you do a lot of highway driving) or 2500 miles or less (if you do a lot of short trips).
Never run mazda's recommendation of 5-20 motor oil. They made that recommendation for 1) a fraction of 1mpg increase in fuel economy and 2) better cold startup emissions in the first 5 minutes of running. We are more concerned with overall engine life than either of those, so we want to run a thicker oil that will not break down as quickly. 10-30 is good for winter or cool climates, 10-40 is good for general use, and 15-40 is good for summer or hot climates. The higher the numbers, the thicker the oil when warm and when cold.
When changing oil filter, place a thick rag under the oil filter pedestal on the inside edge that is tilted down. Unscrew the filter slightly and let the rag catch the runoff so you don't spill it all over the engine. Leave the rag in place as you remove the filter, then use the rag to soak up all the remaining oil in the pedestal.
Consider doing the following maintenance every (or every other) oil change: 1) remove MAF from intake tube and spray carb cleaner in/on the sensor, then reinstall. 2) remove spark plugs, clean with a wire brush to remove carbon deposits then spray with carb cleaner, put a drop of oil on the threads and reinstall. 3) check ignition coils while the engine is idling to ensure each is firing, carefully loosen and pull each plug wire off of the coil and listen for popping, then replace it firmly. 4) with the engine off but the key on, tap dance on the brake pedal 20+ times quickly and watch for the oil gauge to sweep right then left. This resets PCM fuel trims. 5) check or clean air filter.
Consider adding 2 cycle oil to the gas to augment the stock oil injection system and extend engine life. To do this simply add 1 ounce of 2 cycle oil for every 1 gallon of gas you pump. Most times your fill up from 1/8 tank will be about 13-14 gallons, so add at least 14 ounces of oil to the tank before pumping the gas (I would add the entire 16 ounce bottle for simplicity and to be generous). The oil and gas will mix themselves in the tank and that is all you have to do. I recommend to carry a 16 ounce bottle in the car and buy a gallon jug to keep at home, refill the small bottle from the large bottle after each fillup.
Change coolant every 3 years. Coolant turns acidic over time and will begin to eat the internals of the iron/aluminum engine block via electrolysis. There is a drain on the bottom of the radiator and another on the bottom driver's side of the block just above the oil pan. Drain both then add 1 gallon of green antifreeze and 1+ gallon of water.