I didn't read the other posts so sorry if someone's already answered the questions...
Firstly, if there are goldfish in your tank, yes, it is starting to cycle.
Your nitrItes are high because the tank isn't cycled yet. As soon as your tank is cycled, ammonia will be 0, ntirIte will be 0 and nitrAte should be quite high. You are supposed to keep nitrAtes under control by doing 25% (or so) water changes with DE-CHLORINATED water every week. You should aim to keep nitrAtes under 20ppm though up to 40 is ok for most tropicals.
You are lucky to have begun with goldfish as tropicals would be very unlikely to survive the over-stocking, cycling and moving the goldfish have been through. In my opinion, since you have already begun to cycle, keep the goldfish in there until ammonia/nitrite is at 0. I would suggest you do water changes every couple of days though (put de-chlorinator in the water!) so that the goldfish don't have to suffer very high ntirItes.
Then you should get rid of the goldfish like you planned and get your first few tropicals. I'd suggest either 5 zebra danios or a couple of male platies (males have a modified anal fin called a gonopodium - it's tube-like as opposed to that of the female which is fan-shaped). Make sure you return the goldfish and add your tropicals on the same day - otherwise all the good bacteria that has been colonising your filter media and gravel will die of starvation as there will be no fish waste for it to feed on.
The general guideline for stocking a tropical freshwater tank is 'one inch of fish per gallon'. The 'inch' reffers to ADULT sizes so you must know the size the fish will get to when they are fully grown. However, keep in mind that there are certain other things to consider as well - for example, zebra (or leopard) danios need to be in a 20 gallon tank minnimum and in a group of at least 5 because they are schooling fish. Even though they only get to 1.5", you cannot keep 5 danios in a 10 gallon due to their activity level. Similarly, livebearers such as platies breed like rabbits - the general rule is to have at least 2 females for every male or stick to just one sex. You need to keep in mind, however, that females can store sperm for several months so buying only females does not guarantee you won't end up with fry. The reason you keep 2 females to every male is so that the males don't constantly chase one female - which would be very stressful for her. Luckily, most fry get eaten anyway so this isn't too much of a problem most of the time. Platies also get to 1.5" and would get along fine with zebra (or leopard) danios.
Concerning goldfish, high temperatures speed up their metabolism but can also make them vulnerable to certain diseases because they are coldwater fish. Some of the fancier types have, however, been rbed for generations in warmer water and now do better in it. Regardless, the goldfish you have now should be ok in tropical temperatures for a little while until the tank is fully cycled and you are ready for the tropical fish. Make sure you don't have the tropicals in with the goldfish at the same time though - take the goldfish to your LFS or wherever and then immediately (within a few hours) replace them with 2-5 suitable tropicals (as I said, 5 zebra danios and/or 2 male platies are the best starting point). As for the stocking of your goldfish - goldfish get to 12" and are VERY messy fish. The rule for stocking them is 20 gallons for the first goldfish and 10 gallons for each additional tank-mate. That means your 3 goldfish actualy would potentialy need at least a 40 gallon tank. Replacing them with tropicals is a wise descision in that sense and will save them a lot of suffering as they would out-grow the tank and the water quality would suffer greatly.
Welcome to the forum and to tropical fish keeping, don't cut corners, be patient, research all your fish carefuly and have fun!
