That was very good! There are just a few corrections. This would be more natural:

Postcard

Dear Alicia,
I'm very well here. It's a beautiful place, and the weather is nice. I was in many places (It's more natural to say something like, "I have visited a lot of places" or "I have been to a lot of places"). Yesterday, I was on the beach, and there was a big wave. But, I had get back to home because my mom called me and told me that my dog died (or you could say "had died"). I'm very sad, I missed it very much (It's better to say "I miss him/her a lot"), but it's okay. Tomorrow, I'm going to go out with my cousin. It's going to be her birthday. I hope that you are well. And how about you? How's your vacation?
See you later!
Juliana

Postcard

Dear Alicia
I am having a great time here. It's a beautiful place and the weather is nice. I have been to so many places. Yesterday I was on the beach and there was great surf but I had get back to home because my mom called me and told me that my dog had died. I am very sad and miss it very much, but I don't mind. Tomorrow I am going to go out with my cousin for her birthday. How about you? How is your vacation? I hope that you are well.

See you then!
Juliana