Nitrogen cycle lasts from 4-8 weeks, so that's normal... some ammonia did get turned in nitrite and then it should be turned to nintrate if everything is working.
Wait one more week at least before putting fish in tank, check parameters in a week before you go and buy some fish. And don't put too many fish at once. I don't know what you are planning to put in tank but I wouldn't put more than 3-4 fishes at once. Then check your parameters and after 5-7 days put another 3-4 fishes etc. Unitl you are full.
The cycle of a tank is a never ending circle. ammonia=nitrites=nitrates, It is constantly converting. You probably ran your test before the ammonia completely converted through nitrites to nitrates.
Normally we don't see the ammonia or nitrites in a fully cycled and established tank but they are still there. If we have grown a lot of very strong bacteria it does it very quickly so we normally don't see them.
Sorta kinda The cycle actually never ends if one really thinks about it.
But...isn't there always a but
At least once the cycle is stable one should no longer have spikes of one or the other. It should just smoothly go from ammonia to nitrites and finally nitrates that will be removed by water changes or lots and lots of plants.