Tracking Start and End Time of the Chats:
1. Create a Report Type with the Object Chat Transcripts.
2. Add Start Time and End Time while creating the Report.
Agent and Visitor Time Tracking:
1. Create a Report Type with Chat Transcripts as Primary Object and Chat Transcript Events as related object.
2. Create a Report with the above Report Type.
3. Group the Report by Chat Transcript name. Add Chat Transcript Event, Time, Agent: Full Name, Type and Detail fields to the report.