At any train station, 15 minutes is more than enough to change trains. Train stations are much smaller than airports, and you don't have to wait for luggage or to check in.
I suspect (but am not certain) that the train to Milan will use the same platform as the one from Vernazza.
I would say so there are only really two platforms Iin Rapallo going North is platform 1 and South platform 2 . Just wondering g which train you are connecting too as I would have thought that the Milan train would have been stopping in Chiavari the stop before which probably would give you a couple more minutes or if an intercity train possibly Ssstri Levante
When changing in Rapallo, your Intercity train arrives at exactly the same platform as the the Regional trains you have alighted from.
The Trenitalia website is programmed to give the cheapest tickets at the full Base price for the same journey time, which is why it gives a change in Rapallo not in Sestri Levante. At Base price, changing in Sestri Levante is more expensive than changing in Rapallo.
However, when buying reduced advance fares for the Intercity train, changing at Sestri Levante (or even better at Monterosso or Levanto if the Intercity train stops at these) is cheaper as well as giving a longer connection time.
It's not a big station at all. 15 minutes will be plenty. You may not even have to change platforms. There's only two.
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