Ok, I flipped out about a bad answer earlier, and since I can't sleep now (even though I should be) I thought I will write something actually useful. To answer the question

How quickly did they appoint a new leader? How did he obtain the position? Was there mass chaos, or did the transition occur relatively easily?

The chaos developed later. Antony and the assassins reached a deal in a couple of days. None of them had armies yet. The problem came when they got armies some days later. Brutus and Cassius went to Asia, and Antony to Gaul. Decimus Brutus was still in Gaul with his own army, and refused to surrender his command to Antony. By that point Octavian had surfaced and had the mass public appeal to raise his own army, which he did under the order of the Senate. The best evidence for this are Cicero's Philippics, which are both voluminous and online so I won't quote them here. TL:DR, Cicero convinced the Senate to send Octavian and the Consuls to lead an army against Antony. Antony lost the battle at Mutina, but killed both Consuls. Without executive command Octavian arranged joint rule with Antony.

Meanwhile Brutus and Cassius were building their armies in the East, and at Rome Octavian, Antony, and their minor partner Lepidus instituted proscriptions (legal killings on sight) of their political enemies in Rome, including Cicero. That's the first version of nuts. The second version of nuts was Antony and Octavian falling out, which lead to what I think is the greatest naval battle (in terms of ships involved) in history - Actium.

About the treatment of Julius Caesar's death, the Senate was terrified enough in the event to flee the scene, even though only about 10% of Senators were implicated in the crime. Antony and the assassins agreed on rules for the funeral, but the people spontaneously burned Caesar's body in the Forum, which was a spectacular honor. Octavian was also able to raise his own army simply by virtue of being Caesar's heir, so I think it's safe to assume there was a lot of popular support for Caesar even after his dictatorship.

I wrote some more below about conflicts in the Senate about Caesar's position for those interested.

This topic is so vast it makes no sense to me to list specific sources, but if you want to look them up you can look for the lives of Caesar, Antony, and Octavian in Plutarch and Suetonius, as well as the account in Appian's Civil Wars. Forgive me, /u/brigantus.