What type of light do you have on the tank?
How many watts does the light globe have?
How long is the light on for?
What are the tank dimensions?
Live aquarium plants need light and if they don't get enough they won't do well. Depending on how high/ tall the tank is will determine how much light you need.
Assuming there is sufficient light, you can try the following plants: Ambulia, Hygrophila polysperm & H. ruba/ rubra, Ludwigia, Elodia/ Hydrilla, narrow Vallis, common Amazon Sword plant, Water Sprite (Ceratopteris thalictroides/ cornuta).
Water sprite is a floating plant but can also be grown in the substrate. The other plants should be grown in the gravel.
Have the light on for about 12 hours per day. If you get lots of green algae growing on everything then reduce the lighting times by a couple of hours.
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You can order plants online and have them delivered. Sometimes they are good and sometimes bad. Try to avoid getting them shipped when the weather is extremely hot or extremely cold. Plants can rot in hot weather and freeze and die in really cold weather.
Rinse all plants under tap water before adding them to the tank. Check them for small jelly like sacks (snail eggs) and snails, remove any you find.
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For best results you should get some liquid iron based fertiliser for the plants. I prefer Sera Florena but there are other brands available. Get an iron test kit too so you can monitor the iron levels in the water. Add the fertiliser every day or every second day for the first 2 weeks, use a full dose each time. Monitor the iron levels during this time and if the level gets too high, then stop adding fertiliser. Try to keep the levels constant (about 1ppm of iron I think it is, but check the packaging).
The plants will use heaps of fertiliser when you get them but after a couple of weeks of regular feedings they won't use as much and you won't have to feed them as often. But monitor the iron levels for best results.
NB... You don't have to use plant/ iron fertiliser but it does help the plants do better. If you have any concerns about using them, don't bother. Most plants, including the ones I listed above, will do fine without plant fertiliser.
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Different subject now.
How long has the tank been set up for?
Do you know what the general hardness (GH) and pH of the water is?
You can usually find this information on your water supply company's website or by telephoning them. If they don't have it you can take a glass full of tap water to the local petshop and ask them to test it for you. Write the results down in numbers when they do the test.
Neons need to be kept in groups of at least 6 and preferably 10 or more. However, if the GH and pH is too high (GH above 250ppm, pH above 7.8) then I would not get more neons.
If the GH is less than 150ppm and the pH less than 7.0 the platies won't do as well.
If the tank has not been set up for long (less than 2 months) I would not add any more fish until the filters have cycled/ established. You can get the water checked for ammonia, nitrite and nitrate to see if the filters have cycled.