I recently finished cycling my 20G, and I can tell you your tank is not cycled until you get zero ammonia and nitrite. The ammonia you have right now could be from your tap water, so I would recommend you test your tap water so you know what you putting in the tank to start with. My tap water have 0.5-1 PPM ammonia and no nitrite or nitrate. Which was what my water was for the first 2-3 weeks. I did not see nitrite spike until toward the end of my cycle.
If you want your tank to cycle faster, I recommend you try Biodigest. Someone here recommended it to me and I added it to my filter media in the middle of week 3 and it did miracle and my cycle was done within a week. It's a bit expensive but I think it's worth it since i am not very patient and want to get it over asap.
First of all, I add fish food to my tank daily to maintain ammonia at least 1 PPM throughout the process even during nitrite spike. Basically firs the first 2 weeks, I had ammonia at 1 PPM, and no nitrite or nitrate, then I start to see tiny bit of nitrite and nitrate during week 3 but they stay low. And toward end of week 3 I got the biodigest and add it to my water. Pretty much the next day I start to see nitrite and nitrate starting to go up, and after 2-3 days my nitrite was through the roof, it was higher than 5PPM for sure because it's purple and hard to compare against the chart, and my nitrate was going all the up to the 20-40 PPM range as well. And after the 2-3 days of nitrite spike, it literally go away overnight. One morning I wake up and do the water test, ammonia and nitrite was close to zero and nitrate was dark red. Over the next few days I did 25% water change daily to lower the nitrate while monitoring ammonia and nitrite, and they were at zero regardless how much fish food I throw in. After multiple water change to get my nitrate down to 5 PPM, I clean my filter intake (it got lots of fish food stuck to it) and I add another dose of biodigest. Ammonia and nitrite still at zero after 2 days and nitrate went up to between 10-20 PPM. This is when I know my tank is cycled and ready for fish.
Overall, I maintain at least 1 PPM ammonia in my tank from adding fish food, my ammonia never spike, but i definitely see spike in nitrite and nitrate. Not sure if the biodigest help, but everything seem to speed up quickly for me after I added it. When your tank is fully cycle, you will not see any ammonia or nitrite in your tank, only nitrate. I am new to the hobby too and there are still a lot for me to learn here but here is my experience with the cycle.
Bump: Also, forgot to mention I was using the fuval stuff to cycle my tank too, I added it to my tank daily and after 2 weeks I finish half of the bottle and didnt really see much progress that's when I switch to Bio Digest when someone here recommended to me. I still have 28/30 vials of the stuff left and not sure what to do with them now my tank is cycled.