Changing the water will not restart the cycle, only if you change the filter media. Your feeder fish was probably suffering from ammonia poisoning, so changing the water actually helped. Is it a feeder goldfish or a feeder guppy, or something else?
Any filter rated for a 20 gallon tank should be ok, but ideally you want a filter that turns the entire tank over 10 times an hour, so a 200 GPH (gallon per hour) filter would be best. A hang on back filter is probably the easiest/cheapest option for a 20 gallon tank.
As for plants I would recommend Water Wisteria it will help with the filtering duties slightly more than other plants and is fairly hardy. Or duckweed if you're feeling adventurous, but that can take over your tank very easily, so it's probably best not to. If you want a good hardy plant you can't go wrong with some java fern, they do well in any setting and are very very hardy.
For more advanced set-ups you could inject CO2, it can be made from mixing yeast, sugar, and water in an airtight 2 liter coke bottle with some airline tubing running out from it into the tank, or can use a store bought CO2 injector, however those are expensive. For a basic semi-planted tank that's not needed. Generally for plants you want 2-3 watts per gallon, but for java fern and water wisteria you can get away with around 1.