Any of these plants will do well for beginners. http://www.aquariumplants.com/Beginner_Plants_s/17.htm
You want to have some sort of lighting so that your plant will grow. I recommend full spectrum lights designed to give plants the full spectrum of visible light so they have a wide range of light wavelengths to get their energy from.
The plant species I linked you to don’t need CO2 injection (a pain in the butt) or fertilizer (dangerous to your fish). If your goldfish roots it up, just rebury it. These plants are hardy and will do fine.

I especially recommend aponogetons. My first plants were aponogetons; I bought a pack of bulbs from Walmart for three dollars for five to ten bulbs and two of them grew into very beautiful plants with no special care whatsoever.

If you want a plant that your goldfish won’t eat, try this one: http://www.aquariumplants.com/Ludwigia_Broad_Leaf_Ludwigia_repens_p/bp017.htm (read the review that someone wrote)
References :

I have a 30g with a fancy goldfish and the tank is planted. Since your fancy likes to dig try plants like:

Microsorium pteropus (Java fern) Tie roots onto driftwood, rocks, ect.

Anubias nana

Vesicularia dubyana (Java Moss)

Ceratophyllum demersum (Hornwort)

Lemna minor (Duckweed) This is a floating plant, my goldfish treats it like a snack, but it grows back pretty quickly.

I have all of these plants (and some more, but I can’t think of them off the top of my head) and my goldfish & the plants are thriving nicely.

All of the plants I listed are relatively low-light, low-care plants. I do add liquid carbon (Seachem’s Flourish Excel) to all my planted tanks, and it helps the plant growth considerably. You can also add liquid fertalizers, but if added in excess it can cause algae growth. Do you have florecent or incandecent bulbs in your tank? Try to get florcent if you don’t have them. When starting out, try buying a few plants and see how it goes and then gradually build up. Feel free to e-mail me with any more questions.
References :

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