Lots to comment here...starting with the fish. Whomever told you Betta live side by side with tetras in the wild is under a major misapprehension. Tetras are characins, and the characidae live in South America, a couple species in Central America, and a few species in Central Africa. Betta are native to SE Asia; their natural range is throughout central Thailand, occurring in very still and sluggish water such as rice paddies, swamps, ditches, ponds and small streams with very low flow. I do not know what other species inhabit the same watercourses, but I am sure there are some--but certainly not tetras
and in nature fish are able to avoid one another which is something they cannot do in the artificial confines of an aquarium. Now, the Betta has been bred commercially for decades of course, but this does not remove inherent behaviours or requirements, and even to the contrary, the fish has been bred to strengthen its nasty disposition, so they are even more aggressive. You can read more (and very reliable) data here:
http://www.seriouslyfish.com/species/betta-splendens/
To the plants. Light is the most important factor in live plants. Looking at what is in the tank in the photo, the large plant is an
Anubias barteri which is low light. The little plants may be
Eleocharis parvula (Dwarf Hairgrass), or maybe
Lilaeopsis which is very similar. These require bright light. With the light there are also nutrients, and low light means less nutrients than bright light which requires higher nutrients to balance. But if the light is bright, the
Anubias may have problems; algae on this plant is common in brighter lighting. Floating plants would go a long way to fix this. But substrate cover or carpet plants are not the easiest to grow. And another factor comes in here...brighter light is not appreciated by fish. It causes stress which may lead to paler colours.
I might offer some ideas here. With the
Anubias, add some lower-light plants like Java Fern, Java Moss, and/or pygmy chain swords. The latter is South American in origin, but if you are not wanting to be geographically accurate it won't matter. I have this in several tanks, under my low/moderate and moderate lighting.
Not knowing what the present lighting is, in terms of intensity and spectrum, I can't offer much more. But just be cognizant that with higher light comes the need for floaters (for the sake of the fish), and increased nutrients (fertilizers). As it stands now, if you are not using any fertilizer, it might help to consider a comprehensive one like Seachem's
Flourish Comprehensive Supplement. They make several different products under the Flourish name, but this is the complete basic you want. Also keep in mind that as you increase plants, there is a greater drain on the existing nutrients, so if for example they were sufficient now (naturally occurring, without additives) they might not be sufficient with more plants. Especially floating which being faster growing require more "food" to be healthy.
I also see some algae issues in the photo, on the Anubias leaves and among the Hairgrass. This is due to lighting and nutrients, but without knowing more of both it is difficult to pin anything down.
Byron.