The most direct route from Rome to Padua is through the interior, therefore whichever beach you will choose it will entail some detour.
Assuming you don’t have a car, these are the locations with sandy beaches that are close to a train station (walking distance in fact):
Tyrrhenian Sea between Rome and Pisa (STATIONS near beach listed from Rome north).
SANTA MARINELLA
FOLLONICA*
SAN VINCENZO
CASTIGLIONCELLO*
QUERCIANELLA
Adriatic Sea (north of Ancona up to Rimini)
MARZOCCA
SENIGALLIA**
MAROTTA
FANO*
PESARO**
CATTOLICA*
RICCIONE**
RIMINI**
Adriatic Sea between Rimini and Ravenna
IGEA MARINA
GATTEO A MARE
CESENATICO**
CERVIA*
All of the above have decent sandy beaches. Nothing compared to the outstanding beaches of Sardinia, Puglia, Sicily, etc. but those would be too out of the way for your needs.
I marked with one or more asterisks the locations that may still have some vibrant life in October, which is after the beach season is over in Italy.