Moving an aquarium is one of the most stressful things an aquarist (or fish) will ever go through. It is not easy, but if done properly it can go well without a single loss.
Distance and time of the actual move is a major factor. Moving across town or two hours away is completely different than moving across the country. I have moved tanks for myself and clients, and either way it is stressful and a lot of work.
Your best friend in all this is five gallon buckets with lids. You will probably want as many as you can get your hands on. If they were used for anything (like paint, cleaners, etc.) you may want to pass. Hardware stores sell them and most saltwater salt mixes can come in buckets. No matter where you get them make sure they have a tight fitting lid and are clean. If you have any doubts about how clean they are put your face into them and take a big, deep whiff. If you smell anything besides water you probably need to pass on that bucket. Brand new buckets and lids run $5-7 or so at hardware stores, which is more than worth it to make sure you have enough to not crowd any fish during the move.
Stop feeding before the actual move. If you have big fish that don’t eat often like full grown large cichlids such as oscars or big catfish then I wouldn’t feed them for a week. Unless they start fighting each other, just let them go hungry. For most fish 2-3 days should be fine. You want their digestive system pretty empty so that they aren’t polluting their water. This will also let you feed them within a few hours after they are in their new home to see how well they are settling in (they should be happy to eat pretty quickly after getting back in the tank).
Preparing for the Move
If you are worried about water quality then you may want to get some ammo chips. This is the only time I would recommend ammo chips (in filters the bacteria will do it for free). Just put a little in the bottom of each bucket.
In general you will want everything packed up and ready to move before you touch the fish tank. In all likelihood you will be breaking down the tank as the moving truck is being loaded. It will be the last thing you do at the old home and the first thing you do at the new home.
The Actual Move
For saltwater you will want to save as much water as possible. You will need to do this anyway since the best way to transport the live rock is submerged. A five gallon bucket 1/2-2/3 full of water and rock is about the most you will be able to move around. This also leaves a decent amount of air so that there is oxygen.
In freshwater you can worry less about water. Unless your tank only ever gets smaller water changes (such as 25%) then saving as little as 20% of the water will just simulate a nice big water change to start them off well in their new home.
Take out the substrate. Do not move tanks with ANYTHING in them. This is the best way to cause a leak somewhere down the road. Just half fill some buckets with the substrate. Lids are ideal, but if you are short on lids just make sure the substrate buckets won’t tip over.
Buckets with fish should not be full. There is very little oxygen in water, so they need a good amount of air to provide oxygen. I try not to fill more than about 60% water. This also makes carrying the buckets easier. Bigger fish should be separate from smaller fish. For example, my goldfish live with danios, but in a five gallon bucket the danios will be stressed and won’t be able to get away from the goldfish if any decided they were in the mood for a snack.
Except for nocturnal fish mentioned below, there should be no decor in with the fish. They would rather just sit in the bucket, not have to also worry about some decoration smashing them every time the truck turns.
If you have any noctural fish or hiders (like loaches, catfish, etc.) you may want to provide something for them to hide in. Ideally these are light so there is much less risk of them being crushed. So PVC pipes, fake logs, cichlid stones, etc. would all be fine, but actual rocks should be avoided.
Don’t hassle them. Don’t keep checking on them. There really isn’t much you can do, and animals transport better in smaller, dark places. You peeking in on them isn’t going to help you or them.
When you get there you can open the lids to let fresh air in. Take a quick peek to make sure there is nothing you need to address immediately (like a fish on its side that needs an air stone or some fresh water immediately), otherwise just set the lids on the buckets so they aren’t sealed. This keeps them calm and relaxed in the dark but lets some fresh air in.
Get the tank setup and start filling. Start acclimating as you start filling. This won’t be perfect so don’t freak out. It is better to get the fish in sooner rather than later so that if the water filling is different than their temp they go through the change slowly as you fill, not all at once after you are done.
For exceptionally long moves (more than one day of driving) you will want to go the extra mile and run air pumps and possibly even heaters. There are battery powered air pumps (not the worst thing to have on hand anyways in case of power outages). There are also power inverters that plug in to cigarette lighters and provide normal household outlets for air pumps and heaters.