A couple of weeks later, I have decided to tell you how I did it. It's been over a month since the transfer.
The other two answers were good, but they simply didn't work in my case. Since the larger tank was over 5 times bigger, keeping old water wasn't really an option. You need movement in the water so it doesn't become stagnant, especially over a 2 week period. Also since I wasn't just moving the fish from one tank to the other, it was harder to fill up the 75gal with tank water. They were in 2 different apartments, which made keeping water impractical.
Don't get me wrong, if it's practical to dump old tank water in the larger tank to fill it up partially at least, go ahead and do it. But if you are in a situation like me, where it is not practical, filling with tap water is correct. Since most of the good bacteria live in the filter and on other solid objects (substrate, decoration, plants, etc.), by transferring the filter, I had transferred the most of the bacteria.
Here's how I did the move. I had 20-40 red cherry shrimp and ~10 neon tetra, 4 danios zebra and 2 endler. I first removed most of the decoration from the tank. This was mostly plants, a couple of rocks and some wood. Then I started to drain the tank. I took the filter media and dropped them in a large ziplock filled with old tank water. Than I filled a plastic container with about 2-3 gallons of water. The rest went to the sink. When the tank had under 25% water left, I started to catch fish with a net. I put them in the plastic bin for transportation. It's the same bin I dropped 2-3 gal of old water in it. When I had all the fish in the bin, I started to catch shrimps. Those I put in a small bottle filled at 50% with tank water. When every one was in its transport container, I took off. The first thing I did at my new home was to drop the containers in the new tank just like I would have done if I had purchased the fish. While the water was matching temperature, I reinstalled my filter on the new tank. After about 20 min, I started to add water from the new tank in the container. I repeated the above step a couple of times and after that, I scooped the fish out of the container and put them in their new home. For the shrimp I removed the cap and let the bottle sink in the tank (I didn't mind ~500ml of old water in 75gal).
A couple of days before the move, I started to feed less. About 25% less by feeding and the two days before the move I didn't feed. When they were moved, I didn't feed the same day. But the next day I fed them lightly.
It's been over a month since the move and I didn't lose any fish. They are all going great. For the shrimp I can't say because it's too hard, but I still see plenty so at least a large portion survived.