I had a lip piercing that I had to take out after three years due to how it was rubbing my gums, and there is a tiny dimple where the piercing used to be. Nose piercings that are taken out after they heal often leave a similar tiny hole when they are taken out.
At the age of 2 and a half weeks, it is too soon for you to swim in chlorinated water with your new nose piercing. At that time, the outside might seem healed, but the inside will not be. Chlorine burns the tissues, which often sets off irritation. Also, public pools and water have a high bacterial content, and that can cause infection. I would not recommend a tanning bed, as there is no way to tell how the new tissue around your nose piercing will respond. Ultraviolet exposure is usually bad for actively healing skin tissue.
If you need more healing info, there are a couple things for you. One is the article I wrote,
Nose Piercings II: Healing and Aftercare and the second would be the eBook I have available that is about nothing but
Nose Piercings