Quite a lot happened! Here's a rundown of the main events.

First, I should say we have no way of knowing how many people survived the eruption. Knowing how many people lived in Pompeii in the first instance is very tricky and estimates vary widely; then, we would need a way of understanding who was able to get out, so the numbers are really impossible to recreate, and the ancient sources do not provide this information. Probably plenty of people did survive - some probably left in the days leading up to the eruption, if possible, as we have reports of animals acting strangely, springs drying up, things that we know today are signs of major impending seismic activity. Also, the eruption itself happened over the course of ca. 18 hours, and the first 12 of those were ash and pumice fall (note: this is only re: Pompeii itself - the eruption happened quite differently in Herculaneum and in other cities and towns, depending on their position relating to and distance from the volcano). If one was able to get out in the first few hours of the ash fall - most likely by catching a boat or by traveling east, though one could conceivably have gone south if they'd gone early enough and traveled quickly - then they had a good chance of survival.

There was response from the capital. The Emperor Titus actually had two (!) major disasters on his hands at the time (and considering this was 2-5 months since Titus had taken the throne, this was pretty rough!); the eruption, and a major fire in the city of Rome. Suetonius (Titus, 8.3) tells us, "he displayed not merely the concern of an emperor but also the deep love of a father, whether by offering messages of sympathy or by giving all the financial help he could. He selected by lot some senators of consular rank to regenerate Campania (note: not Pompeii, but rather the wider region which was affected by the eruption), and allocated the property of those who had died in the eruption and who had no surviving heirs to the renewal of the afflicted towns."

So: what does this mean? Well, the refugees had to be resettled somewhere, as the areas ca. 10km away from the volcano were entirely uninhabitable and would remain so for many years owing to the amount of ash that literally buried these towns, changed coastlines and waterways, and made for a barren landscape. Inscriptions indicate that there was rebuilding in Naples, Salerno, and Sorrento, and it is reasonable to assume some people may have gone there. Recent research by Steve Tuck of Miami University (OH) has traced the family names of people, via graffiti and tombs, who had previously lived in Pompeii and whose names appear in other places after the eruption; he located many of them in coastal Campania, where they likely had pre-existing social networks from which to draw support.

In Pompeii itself, there was activity post-eruption, which has been categorized as salvaging, looting, plundering, and claiming one's own property - in reality, we can often not know that the motivation for individuals digging in the ash was, but we do know it happened. The city itself was not completely buried - the tops of buildings would have stuck out of the ash, so the location of Pompeii would have been clear and it may have been possible for some familiar with the city to know roughly in what place they were digging (the Forum, for example, vs another neighborhood). One motivation was to salvage expensive materials - namely, marble and bronze (see Jean-Pierre Adam 2006 on this, especially). It would likely have been evident to anyone looking to find such materials where to dig - the Forum was clad in a lot of white marble, as well as temples nearby, and this is largely why the Forum pavement is missing today (Dobbins, p. 155, in Dobbins & Foss 2006). Any number of items went missing in the intervening years, including piping and tanks in the water accumulation towers found throughout the site. It was likely used as a materials mine for a very, very long time before the site itself was re-discovered in the 16th century, and 'excavations' under the Bourbon kings of Naples began in the 18th century (when it was then used as an art mine), before what we would recognize as more systematic, scientific excavations began in the 19th century.

Other evidence of people returning to the site is archaeological; for example, some walls have holes cut into them to allow access through them, as opposed to finding a way through the home's hallways and doors. One intriguing inscription left on a wall of the House of N. Popidius Priscus (VII.ii.20) notes "house tunneled through," seemingly an indication by someone who had returned after the eruption and was leaving a note that they'd been in (and cleaned out?) the house. Excavation reports of some of the wealthy homes indicate a lot of disturbance in the eruption debris from looters/salvagers, more so than the smaller homes (Berry, in Dobbins & Foss 2006).

Sources (secondary):

Allison Cooley & MGCL Cooley, 2004. Pompeii: A Sourcebook (Routledge)

John J. Dobbins & Pedar Foss, 2006. The World of Pompeii (Routledge)

Estelle Lazer, 2009. Resurrecting Pompeii (Routledge)

Steven Tuck, 2020. "Harbors of Refuge: Post-Vesuvian Population Shifts in Italian Harbor Communities, Analecta Romana Istituti Danici; Supplementum 53, pp. 63-77.

Sources (primary):

Dio Cassius, Roman History

Suetonius, Titus

Corpus Inscriptionum Latinarum, esp. Vol IV (but see Cooley & Cooley 2004 for the most relevant to this question)

Vesuvius may have triggered history's first recorded incidence of a disaster relief/rescue mission, launched by Pliny the Elder, commander of the Roman fleet on the Bay of Naples and author of an ancient Roman Encylopedia.

Thanks to his adopted nephew and heir Pliny the Younger, we have an account of his attempt to rescue people using part of the Roman fleet.

Pliny's vessel steered toward Stabeae, where his friend, Senator Pomponianus, was trapped. As the ship approached shore cinders and pumice fell from the sky and the sea became rough. The helmsman advised Pliny to turn back, to which Pliny made his famous reply: "Fortune favors the bold; steer to where Pomonianus is."

Pliny found his friend, but the waves were high and the wind was blowing too strongly, so he couldn't evacuate anyone by sea. Instead he started to lead an exodus inland.

Depending on which account you believe, Pliny, who was quite old at the time, either had a heart attack or breathed in hot toxic gases and died of asphyxiation.

I've never been able to find more information about how successful the rest of the fleet was in evacuating people, or exactly how many vessels took part.

Pliny the younger says, "He ordered the large galleys to be launched and set sail. He steered bravely straight for the danger zone that everyone else was leaving in fear and haste, but still kept on noting his observations (probably for his encyclopedia).

Just an obligatory "I love this sub.."

Thanks to all of you contributing historians. It's a treat to lose myself in here for a while.

I asked a similar question regarding trade and survivors and the amazing u/ToldInStone responded with the following:

https://www.reddit.com/r/AskHistorians/comments/mwc0ir/im_a_roman_merchant_sailing_home_to_pompeii/