Neon tetras sometimes aren't good with bettas. Bettas will sometimes pick on the tetras because they sorta look like a small betta. If your betta likes to be agressive, I wouldn't put anything else in with it. I have had a betta that does good with neon tetras, and one that tries to go after them. Just make sure you have a few hiding spots.
Also on the "tank" you have them in, it is waaay to small for any betta. Why would you not only put a betta in something that small, but want MORE fish? BTW goldfish get big. They need around 10 gallons per fish (I heard) because they get everything filthy, and they get big. They also like buddies, therefore you need a decent sized tank.
Oh, if you can't afford a larger tank, some big plastic storage bins from Target, Walmart, etc... will work great. Same with Kritter Keepers that are 2.5 gallons and up. Remember, you WILL need a heater in with your bettas. They need warm water.