The nitrogen cycle is the process where fish poop, waste food, decaying plants etc produce ammonia which is harmful to fish. Bacteria living mostly in the filter media convert this into nitrite, which is slightly less harmful. Another type of bacteria convert nitrite to nitrate which is much less harmful. Generally you remove nitrate with your water changes although some will be taken up by any plants or algae in your tank.
Cycling your tank is what you do to grow sufficient bacteria for the nitrogen cycle to take all your ammonia and resultant nitrite and convert it to nitrate so your fish arent poisoned.
Broadly speaking there are 2 methods.
A fishless cycle establishes safe parameters before you add fish. You replicate the fish waste by adding ammonia, and over a period (typically 4 to 6 weeks) enough bacteria will grow to safely add fish.
A fish in cycle is where you use the fishes waste to grow this bacteria and control the toxic ammonia and nitrite with water changes until the cycle has established enough for the bacteria to do this on its own. New fish keepers often end up doing a fish in cycle through default because they don't know the options or even that cycling is necessary and start to add fish. This is where you are. It can be done safely and is the choice of many experienced fish keepers.
You should be doing water testing daily, and enough water changes to keep your ammonia + nitrite combined below 0.5ppm. If your ammonia + nitrite is above 0.5ppm do a 25% water change. If above 1ppm then a 50% water change or 2 × 25% changes a few hours apart. When you are consistently seeing 0ppm ammonia and nitrite you should be seeing your nitrate rising and can then cut back on water changes to keep your nitrate down (typically below 40ppm nitrate).
You can speed up the process by adding some bottled bacteria like the safe start you used. This might speed up the cycle from a few months down to weeks. Its unlikely to instantly make your aquarium safe for fish as they advertise however. It might not do anything to help at all. These products are a bit hit and miss.
It would be useful to know what your parameters are if you could post them. pH, ammonia, nitrite, and nitrate. You say they are good, which is great news. If you arent seeing nitrate though, that is a sign your tank isnt cycled and you should keep an eye out for ammonia and nitrite and do a water change if they get elevated as mentioned above.
Also, be careful about cleaning filters. The bacteria you need lives in the filter media. Dont clean anything too much, just a periodic rinsing of any sponges and filter material. And make sure to use tank water to do the rinsing. Chlorinated water might kill off the bacteria.
As much as I thought that I had patience, realise that I might have added fish too quickly. 
