The tank will not cycle itself unfortunately! You need to add a source of ammonia (either pure ammonia or fish food) to the tank. The bacteria needs that ammonia to grow, and will start turning the ammonia into nitrites then nitrates. The API test kit will be much better than the test strips, much more accurate. It also tests ammonia, which the strips don’t do. I personally use both, they’re good to have on hand. For cycling, the test kit is a necessity. You
could do what is considered a “silent cycle”. This involves purchasing a lot of plants and adding your axolotl. The plants use the ammonia and nitrates to grow, and will eat up the ammonia and nitrates while your beneficial bacteria grows, keeping it safe for your axolotl.
For a normal cycle,
1. Buy the API master test kit
2. Add some fish food
3. Test the ammonia in the tank the next morning
4. Add enough fish food for the ammonia to read 3 ppm
5. Test the ammonia, nitrites, and nitrates regularly
6. ”feed” the tank the fish food regularly
Your cycle is complete when you are regularly testing the water and the readings show only the nitrates rising. At first, the ammonia will go up, then over time the nitrites will go up, then the nitrates will go up, then the ammonia and nitrates will start to go down. The nitrates will always continue to rise. Over 30 ppm they can hurt your axolotl. Water changes and plants keep the nitrates down