I'm sorry you are dealing with a mess. I would start with physically removing the algae as best you can & make sure to keep up with your water changes. What does your PH and water hardness look like? If you are working with hard water with high PH, some algae thrives in that environment more than some plants.
Now, I'm a novice with plants as well, so I think you are in better hands with Dave KS as far as plant advice, but if you do have harder water with higher PH, you might want to consider Mollies or Endler's Livebearer instead of Neon Tetras, which like neutral PH (6.5 to 7.5). Some Mollies or Endler's Livebearer eat hairline algae. These are hard water / high PH fish.
I myself have softer water and close to neutral PH, so tetras are pretty easy to keep, but I also have algae that popped up as soon as I added plants, but Brown algae, so I bought some nerite snails and they keep it at bay pretty good.
"The Sea, the great unifier, is man's only hope. Now, as never before, the old phrase has literal meaning: We are all in the same boat" -Jacques Yves Cousteau