Gold fish can be quite destructive on plants, I'm not sure they actually eat many of them but they do dig and wallow around a bit. They are also very dirty fish, they poop a lot. If you are new to aquarium keeping and especially a planted aquarium the pothos and duckweed mentioned earlier would be good choices. With plants though light becomes an important factor, both in intensity and duration. Pothos isn't really an aquatic plant but does very well grown hydroponically and is very tolerant of low or varied lighting. Just drill a hole in the aquarium cover and stick a stem of it down into the water, just and inch or so. It will
root and develop into a cool looking mass that will help clean up the mess the fish make. I'm not sure about the lighting requirements of duckweed but the roots hanging down from it are also cool looking and the fish like nibbling at them.
Other, truly aquatic plants vary widely in lighting and other requirements but they have the advantage of releasing oxygen into the water. You can actually see it in a phenomenon called pearling, where tiny bubbles of oxygen form on the leaves but the light and nutrient balance needed for that isn't easy to achieve. It also isn't essential to be able to see it for reaping the benefits.
There are lots of kinds of truly aquatic plants, defined as plants that grow their leaves underwater. Many have to be grown in a substrate of sand, gavel or even soil. They have root systems just like terrestrial plants. They may really be the best for achieving a fully balanced system but I do not recommend for a beginner. A couple good aquatics that grow outside a substrate are hornwort and Java fern, and they are both a little more forgiving on lighting. Hornwort is very common and can be collected from about any pond or lake. It grows in long string with circular rows of leaves, some people call it coon tail. I like the looks of it best anchored to the bottom it doesn't need that, it can grow just floating around freely. Plus any part actually burred will eventually rot so it ends up floating anyway.
Java fern grows a long rhizome and will attach itself to
wood or rocks. That rhizome must also not be buried in the substrate or it will rot. I'd recommend that if you buy some to find a source where you can purchase a plant already attached to a rock or piece of wood cause it is kind of hard to achieve that from scratch.
There are few plants I can
think of other than the hornwort that you might gather in your own area that are likely to thrive, at least not until you experiment and learn about all the different parameters I mentioned above, namely lighting, nutrition and rooting habits.
For health and of your fish and to make it easiest I'd suggest the pothos, although I'm sure there are lots of other plants that could be used similarly. Actually sweet potato vines do very well hydroponically but would require more light than the Pothos.
When it comes to adding fish, keep in mind that gold fish are slow, with big frilly fins that many other fish can't resist nibbling on so go with docile ones such as guppies or platies. Yes they are often called and sold as "tropical" but have done fine for me without a heater and they are both live bearers so it's fun to watch the babies grow up. A big mess of Pothos roots would be a good place for the babies to hide too, cause the parents will eat them.
Some one mentioned Paradise fish and the are gloriously beautiful fish with interesting personalities but they are murderous bastards. They will just eat anything smaller and rip anything bigger to bits. Florida Fag fish are neat little fish too and very docile but they like to eat plants, especially algae.
I haven't changed the water in my aquarium in 20 years, I just replace as needed due to evaporation. A biofilm whose technical name I don't recall right now covers all surfaces except the front and sides glass. There is no heater, no filter other than a small sponge to catch big stuff and no charcoal. Java fern, hornwort are abundant. At one time I did have pothos but noticed my other plants were suffering, it was too aggressive and hogging all the nutrients.