Fish keeping becomes much much easier once you understand the nitrogen cycle!
As for the platies... Platies are livebearers, and they will continue to breed if you have a male and female together in a tank. Livebearers will usually only eat their own young if they are hungry...but if you aren't sure if your tank is currently cycled this may be a problem. Generally it's recommended that if a tank isn't fully cycled it's best to feed your fish very lightly so their waste is kept at a minimum.
If you don't have one yet, it is very helpful to purchase a liquid test kit (The API Master Test Kit is very good). Liquid tests are more accurate than paper strips, and liquid tests give you many more tests per kit than the paper ones anyway. Being able to test your own water not only saves you the headache of traveling to the pet store, but also can help alert you to any problems in your tank.
Now for a quick explanation of the nitrogen cycle and biological filter in your tank.
The filter in your tank not only helps to trap/remove some of the free floating junk in your tank, but is also the home of important bacteria for your tank. The first set of bacteria break down the toxic ammonia that is released in fish waste as well as other decaying matter in the tank. Ammonia is converted to NitrIte by these first bacteriums. NitrIte is also toxic to fish, but there is a second set of bacteria that breaks that down for food, which produces NitrAtes. NitrAtes are far less toxic for fish, and are removed by us via weekly water changes...usually at least 25% of the water volume.
Remember that bacteria are living things, and they need a food source (Ammonia and NitrItes) to survive. This is why having an empty tank for weeks without a food source does not work. "Fishless cycling" is done by adding a pure Ammonia source to the water (instead of fish adding it), which gives the bacteria food and the ability to multiply.
High NitrAtes can indeed kill fish. This is why water changes, and properly cleaning the tank/filter is important. Remember to "gravel vac" the tank at least once per week. This helps prevent fish waste, and leftover food from decaying in the tank which will eventually be converted into more nitrates! To clean your filter just gently "swish and squeeze" to remove any debris. Do not replace a filter unless it's falling apart. It's not your tank that is cycled...it's actually the filter, so replacing it actually uncycles the tank!
Try to keep below these levels (do water changes as often as nessesary)
Ammonia < .25 ppm
NitrItes < .25 ppm
NitrAtes < 40 ppm (be aware that many people have a certain amount of NitrAtes in their tap water, I have ~15 so I try to make sure my tank stays below 25... test your tap water for NitrAtes so you know how many should be "normal")