What are the tank dimensions (length x width x height)?
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What is the GH (general hardness), KH (carbonate hardness) and pH of your water supply?
This information can usually be obtained from your water supply company's website or by telephoning them. If they can't help you, take a glass full of tap water to the local pet shop and get them to test it for you. Write the results down (in numbers) when they do the tests. And ask them what the results are in (eg: ppm, dGH, or something else).
Depending on what the GH of your water is, will determine what fish you should keep.
Angelfish, most tetras, barbs, Bettas, gouramis, rasbora, Corydoras and small species of suckermouth catfish all occur in soft water (GH below 150ppm) and a pH below 7.0.
Livebearers (guppies, platies, swordtails, mollies), rainbowfish and goldfish occur in medium hard water with a GH around 200-250ppm and a pH above 7.0.
If you have very hard water (GH above 300ppm) then look at African Rift Lake cichlids, or use distilled or reverse osmosis water to reduce the GH and keep fishes from softer water.
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I wouldn't keep big rainbowfish with narrow tetras like cardinals because rainbowfish will eat small narrow fish. I have seen a group of 2 inch long Melanotaenia boesemani eating 1 inch long cardinal tetras.
Virtually all rainbowfish do best in water with a GH above 180ppm and a pH above 7.0. The exceptions are Iriatherina werneri and Rhadinocentrus ornatus. These 2 species are regularly found in soft acid water.
The following link has all the known rainbowfishes from Australia and New Guinea and might interest you.
A website dedicated to Rainbowfishes of Australia and New Guinea
rainbowfish.angfaqld.org.au
If you do keep rainbowfish, make sure at least half their diet is plant based.