Taxis are just outside the station lined up at the taxi stand. They are white and have a taxi sign on the roof. Very easy to spot. Most are regular large sedans that can easily transport 3 or 4 passengers. There is usually a supplement for over 4 passengers (a taxi van will be required in that case). It works exactly like in the US.
There are several stations in Genova, the two main stations are Brignole and Piazza Principe. Trains will stop at both. Piazza Principe is closer to the airport, therefore get off at that one.
There is also the shuttle bus from Piazza Principe, it's called "Volabus", in case you prefer it to the taxi. It's 6euro and takes 30 min.
https://www.amt.genova.it/amt/trasporto-multimodale/volabus/
No Uber in Italy, sorry. The taxi lobby managed to kill it, except for Uber Black (a luxury black limo service operated by professional licensed limo operators) which however is available in Rome and Milan only.