A 'spike' is basicaly the uppermost value it'll reach before coming down again. It varies from tank to tank and depends both on how quickly the bacteria multiply in your tank and on how much ammonia is added in the first place. Like houndour said, it's quite sudden and often goes off the charts. NitrIte also happens to take longest to start coming down again but it's better when you are fishless cycling.
At the moment, it appears you either have a faulty test kit, chloramines in your tap water or have got something organic in your tank as you shouldn't have readings for either ammonia or ntirIte.
First of all, is this water from your tank? If so, you either have plants in your tank or snails (or both) that have already started off the cycle.
Another possibility is that your tap water contains chloramines. Your de-chlorinator would then remove the chlorine but the ammonia (which makes up the remaining part of 'chloramine') was left and that has started your cycle. If it's because of either of these, it doesn't matter and will just mean you have a speedier cycle. Still, planting the tank is an idea to helpr emove the excess ammonia which you'll actualy eb adding at every water change unless you get a de-chloraminator (which is advisable).
However, if this is tap water (not yet de-chlorinated), and you use test strips (the paper ones) to test, you may need to buy a new test kit (go for the liquid-based) as the strips lose their accuracy with time and may be giving you false readings.
On the other hand, if they are correct and this water was not de-chlorinated/de-chloraminated, it emans your tap water contains this toxic substances. If so, look into heavily planting your tank once it's cycled.
If you can test some water directly from the tap and some fresh water that's been de-chlorinated, that would help determine whether your cycle has actualy already begun or not so you'll know approx. how long it'll take to complete.