It's easy. Set up the new tank, run it to aerate the water and get the heat up to match the old tank. Then move any rocks or whatever from the old tank into into the new, put the old filter on the new tank (or if it doesn't fit put the media from the old filter into the new one) and transfer the fish. Done. Keep the old filter, if used, running for about two weeks and then you can safely remove it.
You have transfered the fish, you have transfered the bio-filter. Same fish, same bio-filter. The tank will resume as normal. Acclimation is a personal choice, but I've never bothered since my old tank would be well maintained and the same tap water used for both. Don't add any new fish until you are sure everything is stable, and then remember not to add too many at once.