Well, there are plenty of fish care manuals not only at Petco, but at Boarders, Barns & Nobel, and your local library. Hundreds upon thousands of books have been published with all the information you would need to know about how to take care of an aquarium properly, and then some. From fish health and compatiblity, to water quality and chemistry. Trust me, if you have a question about it, it's been answered twenty times over published in a very good aquarium manual. They're actually a lot easier to understand than the instructions on how to program your VCR. Just have patience with yourself and use your brain.
Yes you have to have a filter on your tank. Unless of course you plan to water change 8 gallons of water every day.
Live plants will not be the solution to your problem, only purchasing a new filter will. In fact, sometimes live plants can make it worst. Unless you are quite prepared to go into aquarium gardening full out (there are manuals on that too), I do not reccommend live plants. They'll only wilt and make the tank dirty.
Goldfish -must- have a filter. No bowl for them at all. They will suffocate in a bowl, unless you plan to do water changes on that bowl every day and make sure it is large enough to have at least three gallons of water per inch of goldfish.
Your "goldfish" are probably some species of barb or platy. Most people will call any yellow fish a "goldfish" for lack of knowing what it truely is. I would not worry about it.
Petco sells three filters that are best for a ten gallon tank.
The AquaClear 150 goes for about $27
The Penguine 125 goes for about $27
The Wisper 20 goes for about $15
You can get them cheaper than that if your store is running a PALS sale, which they sometimes do. As far as I know though, there are no PALS on filters currently.
Good luck
~~Colesea