No bettas don't do well with Goldfish. Goldfish need AT LEAST a 20 gallon tank.. preferably 30 gals or more. (Yes even the little ones). They are extremely messy so they need high filtration (which is hard on bettas). They are also cold water fish and as we all know bettas are tropical fish.
The only other fish you can really have with bettas (although I always advise against having any other fish) are tropical fish.. in which case you would feed them all tropical fish pellets. Bottom feeders get sinking algae pellets.
I highly recommend a mystery snail!! They are from the same family as apple snails, but they stay smaller. 3 inches MAX.
My betta looooves to watch her snail. :D It's so cute. I DO NOT recommend having one in a tank smaller then 5 gallons though. The snail may seem small, and you think, ohhh they'll eat anything, fish waste, algae, etc. They DON'T! Snails poop more then they eat. They poop a LOT too. My snail is pretty lazy and he won't bother cleaning all the algae. Probably because he is spoiled and gets Kale, carrots, bananas, apples, and peas on a regular basis! :)
But over all, they usually do great. Although, some bettas WILL nip at them. My Angel has only nipped him once, and it was when he was first put in the tank. I watched them very closely since then and they do great! He can cruise right past her, feelers out and everything and she just looks at him!
Mystery snails and shrimp are the best tankmates IMO. They both need cycled and filtered tanks and mystery snails should be housed with a betta in AT LEAST 5 gallon tanks. Shrimp are ok in 3+ gallons but it must be heavily planted or your betta will probably eat them when they molt.
Like dh said mystery snails POOP... so unless your vacuuming your tank twice a week (once a week for tanks 10+ gallons) they aren't for you. They also need water infused with calcium so you have to watch your pH carefully when adding a source of calcium (I use cuttle fish bone) because adding too much too soon can harm your betta.
Sorry I may have posted this in the wrong thread didn't mean to hijack. :)
I figured out the damn shrimp grow to be like 1"-2" depending on type so getting more then 2 would violate the 1"/gallon rule etc. so looks like I may only get 1 for now leaning towards the cherry shrimp (no ghosts at any of the lfs.
Tank is been up for a week, has a heater 50w stealth pro (I like the LED light) and the daylight eclipse flourecent lamp. I was going to go buy a plant, havent decided on the kind yet. Heard they help with the cycle.
You don't want to add shrimp until your tank is fully cycled and planted. Shrimp are very sensitive to the water conditions and shouldn't be added to a cycling tank. Your betta will eat the shrimp if they don't have places to hide (i.e. plant cover). Shrimp have very low bioload so in a 6 gallon you could easily have two or three.