Veronica began to date Logan Echolls at the end of the first season. While she was going out with Leo D'Amato, she kissed Logan in thanks for saving her from a would-be kidnapper (Jonathan Taylor Thomas), who actually turned out to be an ATF agent. Their kiss turned into something more when Logan pulled her into a passionate embrace. They begin to secretly date.

Based on this passage, when do veronica mars and logan get together? Let's give stream of consciousness first:
The relevant information to answer the above question is: Veronica began to date Logan Echolls at the end of the first season.
So, the final answer is the end of the first season.