I am thinking about getting a ten gallon (ca. 38 litres) aquarium so that I can add some Neon Tetras to my tank environment that already includes one Betta fish and nothing more. However, I noticed that the Neon Tetras and the Betta fish eat similar things. I don't want to waste money on an aquarium to hold more fish if it will be too much trouble to restrain my other Betta while the Tetras are eating! Is there a simple solution to feeding tank mates separately?
Just feed them together. Use a quality flake or pellet food and it will be fine for both types of fish.
Also, just mentioning this out there because of the common, low-tech / no-tech, betta tank setups: if this tank isn't filtered and heated, do not put tetras in it. Bettas do not have the same water quality and oxygen saturation requirements as tetras do because they have a significantly different respiratory system, called a labyrinth organ, and do not have the same temperature requirements as tetras. Tetras breath oxygen through gills and not a labyrinth organ, they are warmwater tropical fish and need stable water temperature from around the mid 70's °F minimum up to the mid 90's °F (from around 24 to 35 °C). They can actually be in water in the low 100's °F (around 38 °C) for treating certain illnesses for an indefinite period of time if the water is thoroughly oxygenated. But again, stability is fairly important in tetra health.
I have a set up with 9 tetras, a male Betta and a couple of Kuhli loaches and I just trained my Betta to go to my hand when I have it near the water, so I’m able to keep the tetras and Betta separate when they feed, and the Kuhli just eat the leftovers that fell to the ground and pieces of pleco wafers. I think even if I didn't keep them sort of separate in their tank they would be fine. It all just depends on how you introduce your fish and and the fishes' personalities ^_^.
Tetras are one of the best choices out there to be a betta's tankmates. You shouldn't generally have a problem unless your betta is particularly aggressive.
If you're concerned about him exhibiting territorial behavior, you could put him into a cup and block his view of the tank, rearrange all his furniture (maybe even swapping out one or two pieces), add the tetras to the tank and let them get comfortable for a few hours, then finally put your betta back in. With everything rearranged he won't feel so much like he's returning to his old stomping grounds, so he'll be less likely to perceive the tetras as invaders.
I actually have a very similar setup since a couple of weeks. I feed my beta small ball and feed the neon flakes (my betta doesn't seems to like the flake so much...). I feed my betta first by giving him one or two balls at the time. Why? Simple! The neon are way faster than the betta and they eat his food before him if I don't do it this way. Sure they eat a couple of his food but that is not my concern, I want to make sure the betta has enough food so he won't chase the neon around (too much) in the rest of the day.
As another answer noted it is better to had the neon first and than the betta. I actually did it reverse and it worked but getting the neon first should help minimizing the loss. In my case I lost 3 neon over the first week. It hard to say if it was my betta or they were too much stressed by the trip from the pet shop to my home.
To give a chance to all your fish, make sure they have plenty of places to hide. And that the water conditions are good at all time (heater, filtration, biweekly water test, etc).
In my case things seems to have settle down and I haven't lost any fish since 2 weeks. If my betta start chasing the neon hard (not one or two strike but many times) I close the lights of the tank and it seem to calm them down. I never had to do this in the day and only do it at night before going to bed.
Good luck with this fun setup and don't worry about your neon for the food but rather about your betta.