you need fish in the aquarium to get the filter established. A tank does not cycle without fish.
Unless you intend to keep the 10 gallon operating, I'd dump most of the old water in the new tank,
all the fish,
rinse the gravel slightly and throw it in,
put in your decorations,
and top off the tank with room temperature dechlorinated water
and fire up the filter without cleaning it after a bit of the cloudiness settles.
If you think that this process will take longer than a half hour, put the dirty filter material in the tank with the old water, to keep the helpful bacteria alive. Make sure you've set the heater properly. If you have an air pump, use it because the smaller filter won't be moving the water in the bigger tank around enough.
That's about it.
Underfeed for a few weeks till the new tank is stabilized.
When you add the new filter, run both for a while, maybe two weeks, till the new one is loaded up with good bacteria. As long as the tank isn't having cloudy periods, you're ok to buy new fish.
If you want to re-set up the 10 gallon. Just do the opposite.
Take some dirty water out of the big tank when you do a water change, put it in the 10, top off the tank with new water, fire up the filter with it's mostly dirty filter material, adjust the heater, and it's good to add fish.
Use dirty tank water to clean your filter material and you'll keep the biology more active in the filter and have more stable water quality. Always remember that small water changes more often are always safer than big water changes. But no water changes is always bad.