If you move the fish, filter and water into the new tank, you should wait a couple of weeks before adding new fish to the 50 gallon. Monitor the ammonia and nitrite levels during this time. There shouldn't be any but there might. Having said this, you should quarantine all new fish (and plants if possible) for a month before adding them to a display tank so unless you have fish that are currently in quarantine, I would get new fish and quarantine them for a month before adding them to the 50 gallon tank.
If you want new fish now, then set up the new 50 gallon tank and fill it with the water, fish & filter from the 20 gallon. Refill the 20 gallon with clean dechlorinated water and let it aerate for 24 hours. After 24 hours, move the old filter back onto the 20 gallon and take half the filter materials and put them in the 50 gallon. Top up both filters with more material, then get some new fish and put them in the quarantine tank (20 gallon) and leave the new fish in the 20 gallon for a month.
As long as the new fish are healthy and have no issues in the quarantine tank, you can add them to the display tank after 1 month. However, if the fish get sick in the quarantine tank, start the 1 month period after they have been successfully treated for any diseases they had.