I recently moved a dozen or so tanks in my fishroom, Fish in a bucket with a heater & airstone works well. Make sure to put something over the top, you don't want any jumpers, and don't feed them for 24 hours before the move. Less food means less waste.
You can take the same fish from one tank along with the filter, and move them to a larger tank. The filter supports the bio load, the tank is a box that holds water.
You can also split the fish & cycled filter media between two tanks. If you divide the bio load, meaning the fish evenly, putting half of them in one tank, and half in another, you can divide the cycled filter media in half and run half in each tank. This is known as cloning. It doesn't have to be an even 50/50 split with fish & media, and it doesn't have to be mathematically perfect. I do this all the time with angel spawns, it works out quite well.
Your tank water holds no nitrifying bacteria in a cycled tank, it needs something to attach to. It is mainly in your filter media, and to a small extent your substrate, as well as on any decorations including the tank itself.