My fish=in cycling knowledge isn't that great i'm afraid but yes if you have put a fish into a tank that hasn't been cycled then your filter will not have the adequate levels of bacteria needed to break down the ammonia caused by the fish's waste etc. As far as I know (hope someone will correct me if wrong) it's vitally important that the Ammonia and Nitrites be kept below 0.25ppm by performing 50% water changes as and when necessary. You will need to be testing your water parameters at least daily to ensure these levels are kept low.
I have added my 1+month old filter into my old tank to help it with cycle!
Ok. There isn't a lot more you can do until you get your testing kit. When you do, i'd recommend you post your water stats either here or on the new tank forum so that people can advise you further. Firstly you will need to test you tap water so you can get a baseline reading. You will test it for Ammonia, Nitrites, Nitrates and PH. You only need to do this test once. Then, you must daily ( some people recommend twice daily) check and post the test results for your tank water. By posting your results on the forum you will have plenty of people on hand to help and advise should you need it. I find it helps reading other peoples cycling threads (fish-in for you not fishless...unless the fish comes out) There you can get lots of answers to the many questions you may still have.