Hi peediedj67, I agree with Livewire88 - it is wonderful that you have started your tank but not added fish and are starting a fishless cycle. The method of using ammonia to get your biofilter up and going is much more controlled and understandable (it's still a tricky thing sometimes!) than just using fish food.
Even with the calculator it can still be a little tricky figuring out how many milliliters or drops of your particular household ammonia to add to your particular tank. In your case the ammonia concentration turned out a little high. If it still tests at 8ppm all you need to do is replace an inch or two of tank water with some dechlorinated tap water and try to get it down to 4 or 5ppm instead. (You don't really want it up around 8ppm as this can encourage the wrong species of bacteria.)
OK, on to your question of how the ammonia goes down and the nitrite goes up. The first thing to understand is that you are at the very beginning of a potentially very slow process that could take anywhere from 3 weeks to more than a month or two. This is just to say that you want your expectations to be in terms of looking for a change over a period of days rather than right away.
In the cycling process you are trying to create a "biofilter" (biological filter.) Bacteria grow in colonies within a sticky film called a biofilm and mutate into a whole range of types within any given film. All the types we want center around two specific species. The first is Nitrosomonas, which we like the call the A-Bacs because they take ammonia and process it in to nitrate(NO2). The second type is Nitrospira, which we call N-Bacs because they process that nitrite(NO2) into a different substance, nitrate(NO3) which we can remove from the tank water with weekly water changes.
In the first phase of fishless cycling what you're watching for is the first evidence that a colony of A-Bacs has started growing in your filter. For the first few days (or even a week or two) there may not be enough A-Bacs to process any of your ammonia that you can see. But all of a sudden (usually it seems sudden, sometimes it's slower) your concentration of 5ppm ammonia will drop to a lower reading (like 3ppm or 2ppm or all the way to zero ppm) and this will be your first sign that the A-Bacs are growing. Once you test at zero ppm you know you have to dose the tank with more ammonia and put it back at 4-5ppm. (But you don't keep it constantly at 5ppm, you just dose it if you get down to zero or very close to that.)
Meanwhile, after a week or so, or after you've seen some dropping ammonia levels, you should start to hope that your second test, the nitrite(NO2) test, will begin to show some NO2 in the water. When this happens it is a further sign that the A-Bacs have taken off and are really processing a lot of ammonia (plus you'll know you've been adding more frequently.) Once the nitrite(NO2) spikes to as high as the test can measure you know you're in the "nitrite spike phase" and at that point a lot of us like to modify our dosing of the tank down a little, to perhaps 2-3ppm just so that not as much ammonia overall is going in to the tank.
Even though there is a lot of NO2 showing up in the nitrite spike stage, there are still N-Bacs beginning to grow in the filter and they will be slowly beginning to process some of that NO2 into nitrate(NO3) and the NO3 will have an impact on our cycling water. What happens once a lot of NO3 starts to get in there is that some of it is acidic and it can use up all the minerals our water has sometimes and force the pH to suddenly drop down in the pH = 6 range. The bacteria don't like this dropping of the pH and will stop reproducing as quickly! Sometimes we have to do a large water change (with conditioner and temp matching) when this happens to keep the fishless cycling process going along.
I hope this gives you a peek at the road ahead. The members here are great and no doubt will keep your questions answered and keep you encouraged. You'll be a much better fishkeeper for having learned how your filter really works!
~~waterdrop~~
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