If I were in your shoes, this is what I would do:
First, set up the new tank somewhere else in your home, as close to the old tank as possible. Do a full leak test on it, and if you'll be adding any new live rock to your new display (which you should) cycle the new rock in the new tank as completely as possible. Also take this time to do any modifications, test-fit your new equipment, and rinse (but not place) your new sand.
Once you're ready to do the swap... first, set aside an entire day for it, and get as much help as possible. Mix up enough water to completely fill the new tank at least 1.25 times. You'll also want to prepare a number of totes and other containers to hold your rock and livestock while you do the swap. A simple powerhead and heater for each container will do wonders to keep your livestock relatively comfortable during the swap.
Once you're ready to move, time is of the essence. Don't rush, but don't take your time either. Remove your new live rock from the new tank and store in totes. Do the same thing to the current tank - you may want to partially drain before removing the rock to make it more difficult for startled fish to accidentally jump during the process.
Wear gloves while moving the rocks; bristle worms or other stinging inverts may come out during the process, you don't want to deal with that while trying to quickly move your tank. Also be aware that fish may be hiding in any caves in the live rock, so move carefully and try to turn the rock over several times to dislodge them before placing the rock in your external container.
After the rocks are out of the aquarium, it should be easier to net your fish and move them out. Once all the fish are out, finish draining the tank and move it out of the way. Leave the sand in place unless you want to smell something absolutely vile.
If you have the manpower, move the new stand and tank separately to the new location. Big glass tanks are very heavy, so if you don't have any friends to help it might be worth hiring movers to help you. Invest in some glass-moving suction handles to make it easier and avoid pinched fingers. For a smaller tank, you can put your stand on furniture movers and (very slowly) slide the tank and stand as a unit to the new location. That's what I did for my Reefer 350. For a 625, I'm not sure that I would risk it. The tank/stand unit is very top-heavy, which makes it unsteady to move. The stands are also not designed with movement and lateral strain in mind, so introducing too much force in that direction has the potential to cause problems.
Once it's in place, level and shim the new tank - if the current stands have leveling feet, then use those to get your tank dialed in before you start adding water. Once in place, re-assemble the plumbing, set up the rockwork (plus any structural support you might be adding), add the sand and fill the tank. Set up your powerheads, set up your filtration equipment, set up your return pump, and turn it all on. Once it looks good and you've confirmed that there are no leaks, put the fish back in and starting figuring out what you'll be stocking in your much-larger new tank.
