If you’re flying in, you’ll arrive at the unfortunately named Christopher Columbus Airport (GOA). It’s an easy 30 minute bus ride to the city center on Volabus that will drop you at the central train station. The website appears to be down, but buses leave regularly from 5:50 am to 12:20 am and you can buy your ticket straight from the driver. You will be dropped at either Brignole or Principe train station, the latter being the main station.
Check where your accommodation is located but from either, most of the downtown is walkable if you don’t have a lot of luggage.
As mentioned previously, there are two main train stations and many smaller ones for the suburbs. If coming from nearby France or northern Italy the train is usually the best option.
Flixbus also travels regularly from surrounding areas so check timetables and prices to compare.
It is also possible to arrive by sea via cruise ship or ferry. Arrivals and departures include Palermo (Sicily), Barcelona, Tunis, Tangier, Bastia, Olbia and Porto Torres.
Genova offers a free walking tour through the accommodations mentioned above. You will not find a better walking tour anywhere. Spyros is engaging, knowledgeable and hilarious.
If you can help it I would not day trip Cinque Terre and spend a little more time there.
There are amazing day trips to hike, climb and cycle as well as beaches in the summer time.
2 -3 minimum but there is plenty to keep you busy for even a week if you’re using it as a base to visit the rest of the region.