In that case, i would advise that you should take 1/3 of the established media from the old filter and transfer it into the new filter. You can replace it with 1/3 of the new filter media supplied with the new filter.
At the same time, you should add fish to the new tank. The old filter media will be laced with enough bacteria to cope with 1/3 of the waste produced in the old tank, so aim to add the equivalent of about 1/3 of the fish. It is important to add the fish and the mature media at the same time, or it won't work.
By doing this, you can effectively skip the cycling process and have an instantly cycled tank.
So to re-cap
1. Set up the new tank, fill with ornaments, plants etc.
2. Fill with dechlorinated water.
3. Switch on the heater, new filter etc.
Basically your new tank is now running, but without fish.
4. Go to the shop and buy fish approx equivalent to 1/3 of the fish in the old tank. (Do not buy more than this)
5. Bring the fish home and acclimatise them in the new tank.
6. Switch off both filters and swap 1/3 of the old media for 1/3 of the new media.
7. Switch on both filters again and add fish straight away.
Once you have done this, you should monitor ammonia and nitrite levels in both tanks for around 1 week, just in case.
If you detect any ammonia or nitrite in either tank, counter it with large water changes. Ammonia and nitrite are both lethal to fish and levels should be kept as close to 0 as possible. A good level to aim for is to keep both levels below 0.25mg/l, although 0 is best.
Hope this helps. Any more questions, just ask.
Cheers
BTT