In a letter to Tacitus written years after the event, Pliny the Younger described how his uncle, Pliny the Elder, was killed during the eruption of Vesuvius. Pliny the Elder was in command of the Roman navy stationed in the Bay of Naples. When he saw the plume of smoke and ash rising from the mountain across the bay, he initially sent for a single ship, but when he was informed of the scale of the volcanic eruption, he mobilized the entire Roman fleet and sailed to Stabiae, a seaside town a few miles from Pompeii.
Once there, he connected with another Roman naval commander, whose ships were unable to sail out of the harbor due to unfavorable winds. While they ventured down to the shore (with ash and pumice falling heavily around them) to see if any approach to Pompeii was possible, Pliny was apparently overcome and collapsed. Some have theorized that he was killed by noxious gases (Pliny the Younger assumed he suffocated in the ash cloud), but it's not entirely clear what the specific cause of death was.^1
As for that specific rescue operation, it appears to have been unsuccessful. I'm not clear on whether or not any additional operations were mobilized --Emperor Titus organized an extensive relief program, but that was aimed more at recovery than it was a rescue operation.^2
EDIT: I can't speak to how long it took the specific details of the catastrophe to reach Rome, but the sources report that ash fell on cities across Italy (including the capital) and even reached as far away as Alexandria, in Egypt. So people knew something was going on, but not necessarily that it was coming from Vesuvius.