Is there any reason to feed goldfish other food besides actual goldfish pellets? What are the benefits of it?

Just like you and I, they need the other vitamins that some foods just don't carry in them. For a well rounded healthy fish you need to give them the best that they can get

Here's a list of goodies they can eat:

Goldies can eat many different things: lettuce
(darkleaf), spinach, cucumber, zucchini, oranges, apples, grapes,
bananas, watermelon, pineapple, tangerines, peas, dehydrated seaweed,
shrimp, krill, bloodworms, earthworms(put in paper towel for 2 days
to purge the dirt out of their system). Plus they may like some
other things you'd never think of.

i believe the 3 basic and must have (well at least to me ) are;
high quality pellets.
frozen peas.
frozen or freeze-dried bloodworm.

peas can be fed once a week and the bloodworm can be fed every other day.
as for the pellets feed them daily.

as all gf keepers know there are still alot of other food that is great for their gf. but i believe these 3 are my must have.

Goldfish need some variety in their life. It's no fun eating the same food every day is it? FOODS:Give them only as a treat, because uneaten food will polute your aqaurium. Tangerine Sections, Orange Sections, Lime Sections, Grapefruit Sections, Banana Lumps,Peeled Grapes, Washed ( No Peticides) Romaine Letteuce, Boiled and Skinned Green Peas, Chopped Apple Slices, Freeze Dried and Dehydrated Seaweed, Dried Brine Shrimp , Live Brine Shrimp ( Best Newly Hatched), Freeze - Dried Krill , Live Tubidex Worms, Freeze - Dried Tubifex Worms, Live Earthworms ( Only Give if they are found in ground without pesticides, and always wash in cold running water before feeding or put in a paper towel for two days and let the earth come out of it’s system), Finely Shredded , Chedder Cheese, Chopped Shrimp, Chopped Clam, Chopped Lobster, and small Pieces of un - seeded watermelon! All very nutritious and delicious.

Also feeding peas is very good because at least some time in your fishs life it will get constipated. Peas clear their systems.

I noticed my fish were pooping clear poop so i fed them peas and 'WAM' they were back to there normal selfs. (clear poop is a sign of constipation)

Hope this helps

For all of the human food, how small should I cut the pieces? Are the good at breaking things down before they eat it or do they have to gobble everything up whole?

Not sure

i think that most of the time, depending on the softness of the food, you should cut the food up somewhat. for cooked peas and egg white, you can cut them up a bit, and you will see your fish gobble them up whole, or take big bites out of the bigger pieces. i honestly haven't tried any of the other array of foods (like fruits and other vegetables), but i can imagine that you'd use the same method for them. the only issue is when you cut up the food TOO MUCH, then it can cloud your water and be impossible for you fish to eat.
good luck.

If I'm having veggies for supper, I usually serve them to my goldies as well. Last night they had some broccoli, but they didn't like it. My Dojo's attacked the broccoli like it was chocolate or something. haha.

But I've given peas, califlower, brocoli, lettuce, zucchini, eggs, orange, apple, grapes.... all kinds of stuff. Just make sure it's cut up well and soft.

Its fun to watch them eat too.