Sounds like you are planning on moving all your fish to your new tank then and stopping from using the old tank? At least I assume this since you are only going to be running one tank with one filter really......
If so, you could set up your new tank, let it settle a bit, make it the same temperature as your existing tank, and then move your fish and filter over at the same time (your filter must be moved straight from one tank to the other, not allowed to sit around and/or dry out, as the friendly bacteria in it will die if that happens). Taking the water from your old tank would be a very good thing too (discarding whatever amount of water you need to from your new tank).
One of the primary aims of cycling is to get a nice colony of bacteria in your filter which will convert your ammonia to nitrite and nitrite to nitrate... So if you cycle a tank without a filter you won't get this. You will however get bacteria colonies on your gravel, but that most likely won't have alot of filtration capacity, especially if there is very little water circulation (which may be the case if you have no filter and no air stones to move your water around). However... if you did cycle your tank like that, and then put a filter in, the filter would be colonised quite quickly, so you could probably do it. Might take a long time to cycle a tank like this though, especially if your tank is cool because of the lack of a heater. That could slow it down alot of your temperature is low.