Flight
Genoa is the closest airport (but not usually the cheapest option). From here, you can take the bus (either Volabus or the local metro bus) to Piazza Principe and then a train to Camogli. On your way back, if you're flying out of Genoa, you will probably want to stay the night at the airport hotel. Trains can be unpredictable, and traffic can be problematic going by car especially since the bridge collapsed in August.
Flight
Milan is also a good option to fly into and in many cases will be cheaper. Just take the shuttle from the airport to the train station (Milano Centrale) and take a train from "Milano Centrale" to "Camogli-S. Fruttuoso" If you're flying out of Milan, you will want to stay at an airport hotel the night before. Trains can't be trusted in these types of situations.
Travel Note
From the airport, head to the doors on the right side where the buses arrive. This bus will be a regular city bus, marked "SP Staz. Principe" - the service runs every 15 minutes from 06.00 to 20.00 and is free. It is a local bus, and there will be people getting off and on along the way, so it will take longer and will not be as comfortable if you have a lot of luggage with you. Plan for 30-40 minutes to get from the airport to the station.
Travel Note
From the airport, head to the doors on the right side where the buses arrive. This bus will be a coach bus and should be marked "Volabus" somewhere on the front window. It costs 6 EUR per person and you can buy tickets from the driver. You can also buy your tickets online ahead of time for 5 EUR per person, but you must have a printed copy - they will not accept it if it's on your phone. Your luggage will be stowed under the bus, you will have a seat, and the bus will not make any additional stops between the airport and the train station. Plan for 20-30 minutes to get from the airport to the station.
Train
From Genova Piazza Principe or Milano Centrale, catch a train to Camogli. You can buy your tickets at the station from one of the kiosks (make sure you know your PIN if you are paying with a credit card) or ahead of time online. If you purchase a ticket at the kiosk and it's a "regionale", you must stamp it at one of the yellow boxes before you board, otherwise you could be heavily fined. If you buy your ticket online, the ticket is automatically stamped and is good for 75 minutes from the time your train is supposed to leave. So if you miss that train, you can catch the next one. From Piazza Principe, trains run to Camogli roughly every 20 minutes to every hour, depending on the time of day and the trip takes about 40 minutes. From Milano Centrale, you will likely have to change trains in Genova, usually Piazza Principe. The whole journey should take somewhere between 2-3 hours, depending on your exact itinerary. The sooner you can buy these tickets, the better your prices will be.
House Or Rental
Airbnb has some beautiful apartments available to rent in Camogli. Feel free to give us a shout if you want a second opinion on the places you're considering.
Hotel
For a more traditional hotel option that is more upscale than the Airbnb route.