Ammonia and nitrates?

Can someone please explain to me why there's ammonia and nitrates in my tank and no nitrites? The tank has been running for about two months now. It is a 20 gallon tank with 4 danios, and four neon tetras. I feed them as much as they can eat in a minute every other day. I change 20% of the water every other week. The set up came with a filter rated for 20 gallons. What am I doing wrong? I would love to add plants but my office is too cheap to upgrade the lights. What can I do? is my tank cycled (nitrates are present)?