Holy cow, a question that I'm actually qualified to answer!

If someone from modern times were to visit the Colosseum in its heyday, there would likely be aspects that they would recognize from today's professional sports facilities. Many parts of the spectator experience that we take for granted were established by the Romans, and you can see the influence of the Colosseum in things like facility design, facility operation, and crowd control to this very day. During a day at the Colosseum, you would find in-crowd food service provided by hawkers offering options like olives, grapes, and a variety of other fruits from across the Empire. You would also find promotional giveaways and contests involving spectators during the downtime between events, as some of them required setup/cleanup throughout the day.

Other recognizable amenities that you'd find at the Colosseum would include:

• Tickets. The Romans are credited with the first organized ticketing system. Tickets would be inscribed on pottery with assigned entrance gates and seats and distributed prior to an event. The design of the Colosseum and the use of assigned gates for entry and exit led to excellent crowd control, allowing 50,000 people to enter the venue within 15 minutes.
• A variety of seating options. The Colosseum had the equivalent of season tickets in the form of seats inscribed with the name of the owner and box seats for VIPs. It also had had different seating sections for people in certain areas of society (Vestal virgins, religious officials, soldiers, students, etc.), much like we have home/away sections, supporters sections, student sections, or fraternity/sorority seating at different events today.
• A retractable roof made up of cloth canopies that could be unraveled to shade the seating areas
• Elevators for back of house work, like moving animals or competitors from the basement to the arena floor
• The ability to convert the arena floor for different types of events (such as a naval event), much like arenas can convert from sport to sport today.

As far as fan support goes, I can't speak specifically to the Colosseum beyond the seating sections, but chariot races in both Rome and Constantinople definitely did. The major teams were the Blues, Greens, Reds, and Whites (the Reds and Whites were eventually absorbed into the Greens and Blues, respectively), and fans would wear clothing associated with their favorite team as they cheered them on. Team support generally broke down along political, religious, or class lines, and were of particular social importance in Constantinople. Fights (occasionally escalating into full-scale, city-wide riots) were known to break out between the different supporters groups from time to time, which has an echo today in hooliganism.

And speaking of chariot races, the Circus Maximus in Rome had other recognizable amenities of today, including seatback rentals, concession sales, and merchandise sales. Admission to events at the Circus was free, which meant that any revenue could only be derived from those ancillary sources. But seeing as the Circus was capable of seating somewhere between 150,000-300,000, there were plenty of opportunities to make non-ticketing revenue.

Sources:

Fried, Gil. Managing Sports Facilities, 3rd Ed. Human Kinetics Publishers, 2015.

Cameron, Alan. Circus Factions: Blues and Greens at Rome and Byzantium. Clarendon Press, 1976.