Once the balance is reached, a tank will mostly maintain itself. Once in a while, pull out the filters and rinse them in water of a temp similar to the tank-- don't sanitize them, just clear them of debris and ease the over-population of the bacteria there. If you rinse them in a bucket, the resulting brown water is perfect for plants, loaded with the minerals a garden needs.
If you get algae, add a little algae-killer to the water and / or get an animal that will eat it like a snail or a plecostomous. Snails tend to come pregnant, though, and you might wind up with hundreds of snails, and plecos are slow and tend to be inactive during the day. If you can keep the tank out of direct sunlight, the algae should die off on it's own. There are devices you can get to scrape the algae off the glass, and the fish might eat it out of the water, but free-floating algae will just settle and grow somewhere else.
No more than once a month, you can change out ten or twenty percent of the water, but be sure to add more conditioner when you do; most of the time in an airconditioned house, the evaporation will be about as much as you'd want to change, so you just have to top it off rather than changing it, unless it's gotten really acidic.
Periodically check the water's composition with a test kit, or bring a sample to a pet store's fish area for them to test it; make the necessary adjustments to keep it in the ranges recommended for your fish.
A happy tank will start breeding. Look up how to handle it, if you want to keep the babies (for instance: guppies will eat the babies before they have a chance to get big enough to survive, and goldfish require something to lay their eggs in like grass or a breeding mop, and might need a secondary tank for the babies if they're to survive). Frequently, fry need different food than adults, and are too small for the current a filter creates. Also, they'll need places to hide while they're small and edible.
Cats will sometimes try to go fishing, and even if they don't, they're fond of moving water and may drink up gallons in a week. Keep that in mind.
If the substrate gets too clogged with debris, you can get a tank vacuum to clean it out, or can invest in shrimp or bottom-feeders to clean it up, but keep in mind that they'll create waste of their own that will need to be handled.
Chunks of activated charcoal will serve as passive filters-- but make sure it's the kind for fishtanks, usually bamboo charcoal, and not the kind for grills, which will have accellerants that can poison your tank!
Zeolite will counter ammonia, and can be added in small amounts to do so. Large shells will dissolve their calcium into the tank slowly, countering acidity.
Driftwood can offer a nice place for plants and fish alike, but if it starts decaying, it'll only add to your tank-balance problems.