There are two possible reasons that you can't find a "Beam" icon in your Fish Tycoon game. The first is that beaming is supported only in the Palm version of Fish Tycoon, and if you're running the Pocket PC version you won't be able to beam fish. The second is that you don't have v1.1 of the Palm version (if you're running on a Palm device). Version 1.1 can be downloaded from this page.

You have found less than 1/4 of the fish species possible in Fish Tycoon, so there are a lot of fish species left to discover. It can be difficult to find the Magic Fish when you use a random approach to breeding. When you keep the same fish in your tanks, you tend to breed the same fish over and over. It can be difficult to choose which fish to keep and which fish to sell, but there's no way to keep enough fish in your tanks to allow you to breed all of the combinations at the same time. There are just too many fish!

One thing that might help you is an approach that will minimize the amount of duplication in your breeding program. Keep 20 different fish (10 in Tank 1 and 10 in Tank 3), pick one fish to use as a "master breeder" fish and keep it in Tank 2. Breed your master breeder fish with each of the other fish and let the babies grow up. Sell the original fish that you had in tanks 1 & 3 (it's easier if you do it before the babies mature) and repeat the process with the new offspring and your original master breeder. Keep doing that until you've bred several generations using that method. Then pick a different "master breeder" and repeat the process. You'll start seeing lots of new fish (and probably some Magic Fish) soon after you try that approach.