A tank is cycled when ammonia = 0 and nitrite = 0. Nitrate may be showing but nitrate is not as dangerous as ammonia and nitrite. However, you should take care to have nitrate as low as possible too. Too high a nitrate may be bad as well. Try not to have nitrate higher than 20, and having it at zero or very low is best. If you put fish in your tank while you have even small amounts of ammonia or nitrite, they may get sick and die. Ammonia and nitrite are very dangerous and they kill fast. This is why it's very important to cycle your tank before you put any fish in it.
There are various methods of cycling. You can cycle with fish, but remember that if you do, the fish will suffer because of ammonia and nitrite, and they will most likely die. You can cycle fishless by adding fish food to your tank, or by adding pure ammonia to it (but I've never used pure ammonia so I can't help you with it). Just put a few flakes of fish food to your tank every day (and remove it the next day). Keep doing it until your tank is cycled.
When it is cycled, you should start stocking your tank with fish very slowly. Never add too many fish at once because this may suddenly increase ammonia or nitrite, and kill all your fish. Buy 1 - 2 fish at a time, every week or so, until your tank is stocked. Of course, remember not to overstock: do not get fish that grow too large for a 20 gallon tank, and do not get way too many fish as for a 20 gallon tank.
Good luck
