Duckweed is a good plant to grow in tanks with rainbowfish. The rainbows eat it as part of their diet and it helps fight off diseases.
Rainbowfish will eat small shrimp so have lots of Java Moss in the tank for the shrimp to hide in.
Some good plants to try include Ambulia, Hygrophila polysperma, H. ruba/ rubra, Elodia (during summer, but don't buy it in winter because it falls apart), Hydrilla, common Amazon sword plant, narrow Vallis, Water Sprite (Ceratopteris thalictroides/ cornuta).
The Water Sprite normally floats on the surface but can also be planted in the substrate. The other plants should be planted in the gravel.
You can make plastic loops out of airline and a joiner and put the Water Sprite or Duckweed into the loop. It will help keep them separate.
If you add an iron based aquarium plant fertiliser, it will help most aquarium plants do well. The liquid iron based fertilisers tend to be better than the tablet forms, although you can push the tablets under the roots of plants and that works well.
I use Sera Florena liquid plant fertiliser but there are other brands too.
Do not bother adding carbon fertiliser to the tank because they are not necessary.
------------------------
If you post a picture of the tank and rainbowfish, I should be able to ID the fish and we can offer ideas on the tank.