Tap Fish is a pretty straight forward game, but the extensive lists of various fish eggs, in completely random order that doesn't remotely make sense, can be overwhelming. The categories are sort of strange, too. But "starter fish" are indeed a good way to start.
A fast way to gain is to drop each egg in the tank and feed it right afterwards, one by one, because every time your fish are fed you get 1 coin and 1 xp. Every three hours you can hit the "love" button which will get you 25 coins and 2 xp. In addition, you can make the same amount but tapping on the tank and it will reward you just for coming back. And make sure to clean your tank every time it gets green to earn 5 coins and 1 xp.
Once they're grown up, you want to sell them. You can choose between selling them by the bulk via the "Sell" button and selling them individually by tapping on them and selecting "sell". If you sell all at once, you will pay 15% commission for that convenience. So selling them individually will make you more coins.
If you manage to sell the fish when it's very happy, you can get up to a 20% bonus. For that it needs to be over 90% happy. So you want to make sure it has just been fed & loved. Plants and decorations will get happiness up as well. Get cheap ones in the beginning, you can add more beautiful ones soon once you made more coins. If you don't have enough money, you can move the plants in between tanks via inventory to increase fish happiness before selling. The cheapest one to raise happiness is the starfish (store > decorations > tank basics, 50 coins). You'll need a whole bunch to get happiness up. Don't worry if you don't get it up to 91% initially. 81% happiness will already give you a 10% bonus. (I'm adding the 1 because apparently if you only do 90%, tap fish might only give you a 10% bonus because it might think it's only at 89.9 because of hunger or other factors continually changing.)
My personal 2 cents are, the one by one feeding might be good in the very beginning, but it will most likely drive you nuts after a while. Except if you are REALLY trying to save up to something and are willing to put in the effort.
Pink Skunk Clown in my collector's tank. This tank doesn't have plants or decorations because I don't plan on selling from it. So the Pink Skunk Clown is only 81% happy, showing a 10% bonus in the current selling price.
After starter fish, you want to graduate to gold fish pretty quickly. Those are hidden in the "Super Exotic Fish" group. They cost 14 coins and make 12 coins per 9 hours (sell for 26). Go try to fill up a couple of tanks with those. That's for money making, of course, you can always take it more leisurely and collect some fish you find pretty or cool. I had to have some sharks pretty early on...
After that, the White Grunt, also "super exotic", will get you the most profit. At first glance, it might look like others make more, but for quite a while you'll do best with the White Grunt because it is relatively inexpensive for what it makes. You can buy so many more of it for your money, that it's completely worth it. I costs 4,200 and earns 1,680 per day.
Oh, and if you want to decorate but can't afford much yet, you can get some wallpapers that cost 0 coins: Store > Backgrounds > Standard. Also there are some other ones for free: Some colors in Backgrounds > Colored and "Fish Life" in Backgrounds > Exotic. Once you buy a background you can use it as many times you like.
One more thing, if you're afraid your fish will die because you won't have access to them for a while, you can move them to inventory. They'll be fine there.
Happiness to you!