Here's some feedback on your sentences:
1. Going to universidad nacional autonoma de México is big for me. I was accepted the last week and I am going to move to mexico city in August, 4th.
Correction: Going to Universidad Nacional Autonoma de México is big for me (OR: is a big deal for me). I was accepted (no "the") last week and I am going to move to Mexico City on August 4th.
- Remember to capitalize the names of the university and city.
- Michael is correct that you should say "on" if you are referring to a specific date in the month.
2. Hope to hear from you "pull up a chair don't be a stranger"
Correction: Hope to hear from you. Don't be a stranger.
- It is more typical to just say "Don't be a stranger" in this case (without "pull up a chair").
- A situation where you might say "pull up a chair" is, for example, if someone comes into the room where you are sitting in and you want them to join you. Then you can say: "Pull up a chair." (Or you might say: "Have a seat" if there is already an empty chair they can sit in at the table).
3. I have listened to Jhon Newman song and I have a one track mind now.
Correction: I listened to a Jhon Newman song and I have a one-track mind now.
4. I used to be angry with my friend because she always cover hers tracks.
Correction: I used to be angry with my friend because she always covered her tracks.
- This is a little confusing, but people will probably guess that you are trying to say that your friend lies a lot and always tries to hide them. Is that what you mean?
5. I woke up late on sunday but I did my activities back on track anyaway.
Correction: I woke up late on Sunday but I got back on track with my activities anyway. (Or Michael's correction is good too!)
- We capitalize the first letter of days of the week (Sunday, Monday, Tuesday ...), unlike Spanish.
I hope that helps! Let me know if anything I said was confusing :)