The easiest, most effective way (biologically speaking) to quickly make a new tank ready for fish, is to use pre-seeded bio-media. There are bacteria-laden products available to emulate seeded media, but they aren't the same as seeding your own. For quick results, remove some of the mature media from your existing filter and place it in a media bag along with some new media. If you have a tank with an overflow box or you use a sump, consider placing this media bag full of your favorite bio-media in it to hasten the media's growth. There are three advantages for utilizing this type of placement of your bag: 1) it's easier to access than putting it inside a filter, 2) it's hidden from view, and 3) in theory, the bacteria in the bag will benefit more quickly from the continuous, natural flow of the water. In a very short time, you'll be able to remove the media from the bag (or leave it in the bag) and place it in the filter(s) of your new tank.
Yes, you could take some mature media from a running filter, add it straight to the new filter without adding any new media, and then add fish immediately. However, depending on the number and size of the fish you add, the little bit of mature media that you robbed from the existing filter may not be enough to support the bio load generated by the new fish. You could easily overwhelm the media's ability to consume the ammonia and nitrite produced.
Below is an 8" x 12" media bag full of floating plastic bio-media. It's inside the corner overflow of a 75g tank. The floating media keeps the bag above the intake pipe. You'll notice the intake pipe utilizes a prefilter sponge. In this case, I could also use sinking media in the bag, if I chose, because the prefilter sponge would prevent the bag from being sucked into the intake tube opening, which would slow or even halt the water intake.
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