Some people who are not experienced don't really know what to look for, so certain symptoms that you want to look for to diagnose problems or potential problems would be the following: you can look and see if the fish are eating as they normally eat. If fish stop eating or a couple of fish don't eat, that may be a sign that something's going on in the tank. Just like people, when we feel sick, we don't usually want to eat. If the fish are reluctant to eat, they're sitting in the corner, maybe they've got a little shimmy or the fins are clamped, those are signs that something is brewing in the aquarium, and time is of the essence. If you see that something's wrong, your fish that are normally very responsive and right at the top of the tank to consume food are now sitting at the bottom of the tank, something's wrong.
It could either be water chemistry, or maybe you've introduced a parasite into the aquarium from adding new fish. Whatever it is, you want to jump on it soon before it becomes a full scale problem. Look at the fish, see if they have any symptoms on them, if there's any body slime, if their eyes are a little bit cloudy, if they have white spots, and then either research these symptoms online further to see what could be wrong, as certain symptoms on the fish can be identified. White slime could be ammonia buildup, red burns on the fish could also be ammonia buildup. White spots on the fish could be a parasite. Cloudy eyes could be an external trematode burrowing into the eye. Nasty stuff, but these are the symptoms you want to look for.
You also always, anytime something seems out of whack, test the water. Make sure your heater is functioning. Sometimes heaters stop working and you don't even realize it. Your thermometer broke months ago but the heater has been working, and your water temperature is too cold. The only way to know this is by looking at the fish and seeing that they're not eating as much, so you do a full-scale test. Test the temperature, test the pH, test the ammonia, test the nitrite. Make sure all your filters are plugged in and they're all working.
Sometimes filters become clogged or they just stop working mysteriously and they go unnoticed, and the tank has been running a couple of days without a filter. The first thing you want to is look at the fish and see how responsive they are. They are like the canaries in the coal mine. If the water gets a little cloudy, that could be a symptom that the tank is getting a little out of whack. Maybe you haven't done a water change in a while. The filter is clogged, and now the bacteria, which can no longer live in the filter because it doesn't have the water flow it needs, it is now growing in the water column, so the tank is getting a little bit murky.
If you see a lot of waste settling on the gravel bed, maybe your filter is clogged and the flow has slowed down to a point where it's not keeping the waste in suspension and now the waste is settling on the bottom. A large buildup of algae would also signify that there are a lot of nutrients available that may have not been available months ago when the tank was working properly.
Again, look at the filtration. Maybe it's slowed down, maybe it's time to clean it. Fish grow, and as they grow, they excrete more waste and they need more food, so the rules that applied last year or six months ago may not apply now. Your fish have doubled in size, and now the tank may be overcrowded where it wasn't overcrowded six months ago. All of these things should be things that you're looking at.