What systems are available today that log entry and exit into a building. Provides reporting capabilities. Currently have a badge system but does not track staff who follow each other in or out. RFID badge system?

RFID badge which stores these details and could be used for tracking and monitoring 

You should consider BRIVO. BRIVO is a compelling choice for a physical access solution due to its approach to security and access control. Its a cloud-based platform which is highly flexible and scalable system that can adapt to the unique needs of businesses of all sizes. It simplifies access management, making it easy to control and monitor who enters and exits your facilities in real-time, regardless of your location. Most importantly BRIVO's integration with various security systems, from video surveillance to intrusion detection, enhancing overall security make ecosystem integration easy.

Where it was required to be very accurate like chemical plants and security buildings we did turnstiles but also had security staff at the entrance
Any entrance that needs tight security still needs humans. Maybe AI changes that some day

We use Kantech system. Badge Swipe with full reporting, and access user level definition, even by time of day.

Tracking groups coming in together and tracking exits is tricky. We used to use an old Keri Door Access system that allowed us to track exits. This was also dependent on if the individual remembered to tap their access card.

I believe Verkada has functionality like this for their access system. If you aren’t looking to replace your door access system, you could utilize a visitor management system. In such a case, you might have an iPad at each entrance, where employees would scan in and out. This is still up to individual compliance, but exits will always be subject to compliance, since you can’t lock an exit door like an entry door, for safety reasons.