A few years ago when I did exactly the same thing as you, upgraded from a 75 to 125 gallon, I siphoned the water for my 75 gallon into a water butt. It didn't quite hold the full amount, but it was quite a lot, you don't necessarily have to use all of the water, but it does help if you use some of it. I had started running a canister filter on the 75 gallon to help with the cycling. I then removed the filter from the 75 gallon and attached it to the water butt so that it kept running. This is very important, you mustn't leave the filter turned off for too long or you run the risk of all the bacteria dying. Try and use as much of the old substrate as you can, although there's not much bacteria contained in it, there is some and it will certainly help with the new tank
Once I have removed the water, it was just a case of moving the tank out of the way, putting the new one in its place, and then putting the water back in. I used a power head to pump it from the water butt back into the new tank.
It's quite a simple job really, obviously the hardest part is moving tanks out of the way if you need to do that.