I swear by the API master kits. They tell you invaluable information. You can snag one at Petco/Petsmart for roughly $35. Or on Amazon for about 28. They are good for hundreds of tests.
As far as the filter, you don't need to change filter media often. Whether it's ceramic, sponge, charcoal, sand, or ready-bought HOB floss pads. You should pretty much keep the same filter media until it falls apart. Rinse as needed in tank water. If you need to replace it, I suggest adding in half new while keeping half the old. This allows the bacteria to colonize the new media before throwing the rest away. You can even put a new corner filter in the tank with the old one and let the new one season before removing the old one.
Regardless, I do think everyone can benefit from a sponge on their intake, unless someone has a fancy sump setup.
Beneficial bacteria are what "cycle" your tank. They live on the decor, substrate and, in large numbers, on your filter media. Some consume ammonia (nitrosomanas) that is created by fish and food waste and produce nitrites as a byproduct. Others consume nitrites (nitrobacter) and create nitrates as a byproduct. Both ammonia and nitrites are harmful to fish. Nitrates are only harmful in massive numbers. Water changes are what remove these nitrates at the end of the cycle.
It takes time to develop enough bacteria to complete the cycle. If you remove too much of the bacteria for the remaining amount to keep up with the waste produced in the tank, ammonia spikes and can cause burns, susceptibility to disease and even death.
There's also the fact that some hardier fish can actually get used to an imbalanced tank. They adapt to dirty, hard, soft, acidic, alkaline, or otherwise unhealthy water. Then, when fresh water is added, it shifts the parameters very quickly and the fish die from the shock.
Hopefully you'll be able to get your hands on an API kit and we can try to figure out what's off in your tank!