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. It was hard for me at the beginning because I ran out of 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. Then after a few attempts I could run 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. Now I can 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. I try to keep a slow pace most of the time, but sometimes I have to quicken at a traffic light 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. Other times I may speed up when I feel like it, if the road is straight and there are no cars.