The first thing you need to do is make sure you have everything you need in advance. The last thing you want is to be half way through and have to dash out to buy something.
Feed the fish less than usual for a few days before and after the move and nothing on the day itself.
If you will be using sand it will need washing. Do this in advance as some sand takes a lot of washing!
Once you are ready, the first thing is to empty the 60 litre tank. You'll need as many buckets/storage boxes as necessary to hold the water in the 60 litre.
Start by transferring water to one bucket, then put the decor in there.
Then fill another, catch the fish and put them in bucket 2. It will be easier to catch them with the decor removed. Put the heater and filter in the fish bucket and turn them on. Cover this bucket as stressed fish will jump.
Continue emptying the tank into buckets until you've got as much water as you can without stirring up muck from the bottom.
Move the old tank to somewhere you can abandon it for a while.
Now move the new tank into position. Put the substrate in followed by a few inches of new, dechlorinated water. If you have live plants, now is the time to plant them. Attach the new heater to the tank.
Now start adding water from the buckets with just water in, then add fresh dechlorinated water warmed to the right temp until the new tank is almost full. Leave enough room for the water in the decor and fish buckets.
Now place the decor and water in the tank and arrange to your satisfaction. Turn ff the heater and filter in the fish bucket and take the filter out. The next step is up to you. You can either run both filters for a couple of months or you can remove all the media from the old filter and put it in the new filter, filling the gaps with new media. To be honest, I'd do the latter. Put the filter(s) in the tank and turn both heater and filter on.
Finally catch the fish and put them in the new tank and the water from that bucket. Strictly speaking, you should bag them in water from the old tank and acclimatise them to the new tank but since you'll be using all the old water it's just like doing a huge water change. As long as the temperature isn't greatly different they should be OK going straight in. Finally top up the water to the full mark.
Leave the lights off for the rest of the day. Check for ammonia and nitrite every day for the next few days. You will lose the bacteria in the biofilm on the glass, the substrate and the old filter casing and heater. There will still be lots in the filter media and on the decor and they should make up the numbers quickly. But it is better to check just in case.
Then get a cup of tea and go and deal with the old tank!