Preliminary notes, still in preparation (2015-Aug-13), available here. I have notes prepared for a previous summer school with a different emphasis here. They might be useful.
I think one can start with this survey; follow the references for Kato's papers.
Regarding log stable maps, the paper of Gross-Siebert is here. The papers of Chen and me are here.
You can also take a look at the papers of Chen and Zhu.