The water scenes in the arena were filmed in a water park south of Atlanta. The water was barely more than 10 degrees Celsius (= 50 degrees Fahrenheit), and Lynn Cohen (Mags) was exempted from going into the waters at the request of the director Francis Lawrence, who was concerned for her health. In the first scene, where she is carried on Sam Claflin's (Finnick's) back, he loses his footing, bangs his knee on the rocks, and dumps them into the water. He felt horrible about it, but she found it hilarious.
(At around one h 4 mins) When Katniss is in the archery simulator, her targets resemble her fellow tributes. Notably, one throws Brutus's spears, one hurls axes like Johanna, and one carries a trident like Finnick.
The book describes the previous two-quarter Quells. For the first, District members had to vote on who would be selected as the tributes, and the second had twice as many tributes, two boys and two girls from each district. The book also reveals that Haymitch won the second Quarter Quell.
(At around one hour 45 mins) When shooting the scene where Katniss, Finnick, and Peeta eat fish, the actors and actresses are eating raw fish. Neither Jennifer Lawrence nor Sam Claflin is particularly fond of fish, making it difficult for both actors to eat it. Lawrence had the most challenging time, and Josh Hutcherson and Claflin found her struggle amusing.
In the first movie, you know Peeta's and Katniss's training scores. The scale is 1-12. Peeta was eight, and Katniss was eleven. In this movie, you don't. In the books, they both get 12s. They are the only tributes ever to get 12s--suggesting it's because the game makers want to punish them and make them targets for the other tributes.