I am confused by these sentences, 

1) Yarın dedelerini ziyaret edecek. 

2) Geçen Pazartesi arabasını sattı.

I am confused by the -ini endings. I thought that -in was used for possesion? It looks like its trying to do the possesion+specific but I dont know why in the context of these sentences. 

Good point ´in´ and ´ini´ can be taken different ways. On their own, they are ambiguous.

Here are a few examples:

This is Tim´s book.

Bu Tim´in kitabı.  Here ´in´ is for the possessor

This is your book.  

Bu kitabın. Here ´in´ is for the object being possessed.

I took his book.

Onun kitabını aldım.        Onun is used to clarify Kitabı and ´n´ is a buffer letter

I took your book.

Senin kitabını aldım. Senin is used to clarify Kitabın, like you said possessive + object

The last i of dedelerini goes with ´ziyaret edecek´, it is specific.

Same goes for arabasını.

In Turkish you actually literally say ´the his car´.

There is some ambiguity here. For example this could also be: Tomorrow he visits your grandparents - dedeler-in-i. (or less likely, their grandfather dede-leri-n-i) There should be some context to clear this up. Otherwise you can write: Yarın onun dedelerini ziyaret edecek to make it clear it is his/her grandparents.