Hi Cossie,
Welcome to the forum. Unfortunately, Cezza is correct. Having the filter running and the heater on does nothing to prepare the tank for fish. All it does is confirm that those parts are working and the tank is leak free. The maturing process of a tank, is really more about the filter and the bacteria that process the wastes that your fish produce. The fish continually will produce ammonia and this is excreted primarily through their gills, in addition to their poo. This ammonia is actually toxic to the fish, just as carbon dioxide produced by humans is toxic to us.
The production of ammonia by fish can be simulated by adding pure ammonia to your tank (perfume and surfactant free). This is known as a fishless cycle. The bacteria that process ammonia into nitrite come along and produce a secondary toxin. The nitrite then will be processed by another bacteria into nitrate, which is FAR less toxic. This is known as the nitrogen cycle, as you start with NH3 (ammonia) then process to NO2 (nitrite) processed to NO3 (nitrate). Nothing can really remove all the nitrates other than you, and so you are required to do maintenance water changes on a regular basis. I do 30% weekly.
The other option would be a fish-in cycle, which is much more difficult and needs to be VERY closely monitored. Both are effective. Generally the use of "instant" cycle products isn't considered to be effective enough to allow you the luxury of just banging your fish in to the tank quickly.
The fishless cycle could take as little as 4 weeks or as long as 10+. But, at the end, you'd be able to add ALL your fish at once, although I've never trusted that and would never go higher than half the stocking at once. The fish in cycle will require countless water changes, stress for you and your fish and you probably won't reach your total stocking levels for 4-6 months. Please do read the link Cezza offered for the beginner's resource center. It is full of LOTS of information. It will probably cause information overload.

Take it slow. Doing a fishless cycle will allow you to read through all the different resources and learn a great deal stress free. Doing a fish in cycle will get you to the same end, but it will likely be far more stressful in the long run.
As far as your stocking list goes, very nice. Some of my favorites. Harlequins are very hardy little fish, as are platies. And personally I just love cories. Depending which method you go with to cycle your tank, what the answer to your question is. Personally, I'd fishless cycle. Then I'd add the whole shoal of harlequins first. Then give it about a week or two to settle in and ensure that your filter is handling everything well. Then, I'd add the platies. Finally, I'd add the cories last. (The cories are the most sensitive of your list.)
That's a slightly heavy stocking load for a new aquarist. Just keep an eye on your water parameters when you reach max load and be prepared to do larger or more frequent or both water changes to keep things all in check. Best wishes with it all.
