Upgrading to a larger tank is a great idea and something you won't regret. Bigger tanks are easier to maintain, not to mention that you can keep more fish. The fish, as well as the gravel, can be transferred, but I'd leave most of the water behind. The water itself isn't the source of bacterial colonies. The beneficial bacteria colonize the filter media, the gravel bed, and other hard surfaces of the tank, such as rocks and decorations.
Here are the steps to successfully transfer fish from an established smaller tank to a larger new one.
Prepare the Old Tank
Lightly vacuum the gravel in the old tank to remove any large particles. Test and record the pH and water temperature so you can match them to the new tank. If the tank has recently gone through a major event such as fish death, replacement of the filter medial replaced or new fish being added, wait a few weeks before undertaking the move. This will allow the old tank to stabilize.
Set Up the New Tank
I assume you'll have a new filter and heater for the larger tank because the old ones won't be sufficient to filter and heat it. Set up the tank and fill it three-quarters full of water. That's enough to set up and start the new filter and heater, and it leaves room for gravel and decorations. Let it run for 24 hours.
If you plan to use the old filter in the new tank, leave it running on the small tank.
If you have one, run an air stone in the new tank. That will help drive off any chloramines that may be in your water supply.
Reconsider your decision if you're planning to use the old heater in the new tank. Even if it isn't too small, the new tank should be run for a while with a heater to ensure that the temperature is stable.
It's also wise to have more than one heater on hand because you never know when one may fail. So get a heater for the new tank, then you can keep the old heater as a backup after you take down the old tank.
Test the Water
Test the pH and the temperature in the new tank after 24 hours and compare the results to the old tank. Don't proceed with the transfer if there is a difference between the two tanks of more than two-tenths in pH or a couple of degrees in water temperature. Of the two parameters, the pH is the most critical. When that matches closely, you are good to go.
Move the Gravel
Use a clean cup to transfer the gravel from the old tank to the new one. I like disposable plastic drinking cups that I can throw away when I'm done. Move any rocks and decorations next. If you plan to use the old filter in the new tank, move it at this time and make sure it's running before moving the fish.
Move the Fish
If the pH and the water temp match exactly, you can transfer the fish without bagging them. Otherwise, you'll have to bag the fish and acclimate them the same way you would with a fish you bring home from a pet shop.
Use a disposable plastic cup and a net for a direct transfer. Hold the cup underwater and use the net to drive the fish into the cup.
Lay the net over the top so the fish can't jump out, then place the cup in the new tank and let the fish swim out.
The cup-and-net technique might not work for all fish, but it's a less traumatic way to move many of them. For those that can't be moved using a cup, net them and place them in a sturdy baggie filled with water from the tank.
Move the Filter
Wait, didn't I advise buying a new filter for the new tank? Yes, I did. But the filter from the old tank has beneficial bacterial colonies in the filter media. By running both the old and the new filter for a few weeks, you'll introduce a lot of beneficial bacteria into the new tank. The extra filtration will also be helpful as the new tank becomes established.
After the Move
You'll still go through a startup cycle, but transferring the gravel will speed up the process.