Imagine the situation, we play a game like ping-pong or badminton or something like this. At the start, we play not counting points (for training e.g.), then one of us proposes: "Let's start counting points". But this phrase sounds awkward to me, what would one say instead? Also is there words for "playing game not counting points"?
The phrase you're looking for is to keep score. You would use it like
Let's keep score
Let's start keeping score
I can't think of a term that exactly means "not keeping score", although you could say "We're not keeping score." The closest term might be a friendly game, although that just usually means that the game doesn't really count for anything or there are no stakes; you might still keep score in a friendly game.
If you were to say "Let's just warm up for a while" your opponent would understand you to mean that you don't want to start the actual game just yet. You would be getting limber, getting the feel of the playing surface, etc. You would not be "keeping score" yet, as standgon correctly says in his answer.