Hey Dave! You sound like your betta has a great home.
Even in a cycled tank you will need to make weekly water changes. We're putting stuff (food) into the closed system, so we'll need to take stuff out (waste and water). Plants help, but they don't eliminate the need for water changes. So once you get it cycled you're going to be looking at one 30%-40% water change weekly. If you want sand (best for your cories) then you will still need to suck the poop off of it during your water changes.
Proper care for any fish is mostly weekly water changes, good food, good water, and proper setup. I would actually go with cardinal tetra over neons because cardinals can handle the higher temps that betta require while neons actually do best around 75F. With either neons or cardinals you are going to want subdued current and floating plants. This will help them feel much more comfortable and be much more brilliant in color. I'm going to warn you, some people have had trouble with neons nipping fins. Ember tetra are another possibility you can look into that others have had success with. Alternatively, you could leave the betta in the 5 gal (where he would certainly still be very happy) and populate the 20 gal with more neons/cardinals and cories. Without the betta you could up the neon school to 10 or 11 and the cory school to 7 or 8. Both of these fish will look more impressive in larger schools. I should also warn you that neons and cardinals are soft water fish and will live longer in soft water. In hard water, they do have shortened life spans. Cories can tolerate harder water than neons and cardinals can.
Hope I answered all your questions. Feel free to ask more! We're here to help!