While remembering the colors and finding ways to play your cards are both important. At it's core, this is actually a disguised game of Nim.
When you start a fresh rainbow, always make sure to play exactly 2 cards. That way, since your opponent has 4 open spots, they will always have to leave 1, 2, or 3 spots open for your next turn. Allowing you to complete the row.
Playing only 1 card would allow your opponent to take back the advantage by also playing a single card, leaving you with 4 open spots. Playing 3 cards on a fresh board would leave your opponent free to try to finish the rainbow on their next turn.
If there are already 2 cards on the table only add one as you would be unable to finish in one go.
If there are already 3 cards on the table attempt to finish in one go.
Take into account what colors are revealed to each player.
- If you play a card flipped, you risk an error or a shadow as you don't know what is on its back, but you gain the advantage of having seen both sides while your opponent has only seen one.
- If you play a card without flipping, you play it safe, but reveal both sides to your opponent while you have only seen one side.
Taking risks at the beginning is better than later as you risk loosing fewer cards. The information you thus gain might help you complete a rainbow and the information you hide might lead your opponent to an error.
Specificaly playing colors that correspond to those you see in your opponent's hand increases their chance to make an error when they later flip those cards.