It was hard for me at the beginning because I ran out of the breath too soon and I had to take breaks and walk.

Then after a few attempts I could raun around the block where I live, and it became easier after some time.

Infinitive with "could". Not sure if the comma is necessary but it feels like this sentence needs a pause somewhere.

Now I couldan run 8-9 kilometers without getting very tired, and I feel that I can do more.

"could" is either for expressing past ability like you used it earlier, or for expressing something that *might* happen. "If I trained harder, I'm sure I could run for ten miles."

Expressing present ability like you have here is not "could", it's "can".

I try to keep thea slow pace most of the time, but sometimes I have to quicken at thea traffic lights while it is green.

Or "at traffic lights while they are green". You can speak about traffic lights in general, or you can speak about a single traffic light that acts as a representative for all of them, but you need to match plural.

I don't know if this is colloquial but in the UK you actually could say "at the traffic lights", but you'd say that if you were thinking of a *particular* junction and your listener probably knows which one you mean. I was running down the road and making good time but I had to stop at the traffic lights (you know, the junction by the pub).

Other times I may speed up when I feel like it, if I the road is straight and there are no cars.