I highly doubt you will have an ammonia/nitrite spike. Just feed lightly for a few days to let the filters build up their beneficial bacteria. Once you have an established colony of bacteria it does not take very long for it too grow as needed. This would be especially true if you just did a few large water changes to remove large amounts of waste products. Just feed lightly for a few days and don't clean your filter for about a week.
In the future, try to find out what the actual nitrate levels are, because high could mean 20 or 80. 20 is higher than a lot of people like it (and too high for some specialty fish), but for tank raised fish it is not a big problem. I just wouldn't let it get much higher than 20. If it is 40+ then yes you need to do some water changes and tank/filter cleaning to remove waste products.
Now someone mentioned that you needed to learn more about the nitrogen cycle, but they didn't explain it and I didn't see that anyone else did - so I will give a simplified explanation.
Fish create waste in the form of ammonia, and any uneaten food or dead fish will increase ammonia production. Ammonia is highly toxic to fish, but in a cycled tank there are beneficial bacteria that convert the ammonia to nitrite, which is also highly toxic. Again, there is another type of bacteria that converts the nitrite to nitrate, which is not toxic unless it gets to high levels, as described above.
The only way to remove this nitrate is to have aquatic plants, a nitrate filter (very expensive and can be difficult to set up), or water changes. Water changes are usually the easiest and cheapest fix, and are necessary no matter what other step you take to remove nitrates. Even in a heavily planted tank that takes care of nitrates you must change some of the water because other nutrients required by the plants are removed over time. There is also debate about hormones that are released by fish that can build up over time as well. So water changes are always necessary, it is just a matter of how often. The last option to removing nitrates is submerging the ROOTS of some terrestrial plants, with pothos being the most popular. Something to remember about plants though is you have to remove the dead/dying leaves. If you have a dead/dying leaf submerged in the water it will release all of the nitrate it has absorbed which can quickly lead to disaster.