Pliny the Elder died during a rescue effort (he was commander of the nearby Roman fleet) and Pliny the Younger wrote a detailed eye witness account of the eruption which is basically the basis for volcanology as I understand.

Suetonius (Titus 8.3) says the emperor Titus arranged for aid for the survivors by a commission of former consuls and rebuilt some cities using the property of the dead who left no heirs.

Cassius Dio (66.24) says Titus at least toured the affected area himself. He also reports many private citizens and cities and client kings offered money for relief of that disaster, but Titus refused the money and paid himself.

So it seems there was both an immediate response on the local level and long term relief administered by the highest office, as well as charitable persons willing to help across the empire, but whose help wasn't accepted.

I have to admit that last bit feels strange to me. So much of our disaster relief is conducted through private organizations - Doctors without Borders, the Red Cross, etc, it's hard to understand why Titus wouldn't want extra money. I'm wondering if it's a patronage issue? Something about Italy's special status vis-a-vis the provinces? I don't really get around much after the Julio-Claudians.