Hello, I'm planning to use Goeuro from Genoa to Pisa. I would like to know if it is a trusted place to buy the ticket, because by comparing prices with treniitalia, this website is cheaper. Thank you
It is not a trusted website. It is a third party agent, which cannot possibly sell tickets cheaper than the train operator Trenitalia, and may (I do not know) also charge an additional fee. Either you are not comparing like with like, or GoEuro does not display correct prices until you start booking a specific train. And if anything goes wrong, it is better to deal with the train operator than a far away agent.
Buy from http://www.trenitalia.com/tcom-en entering Genova Brignole (or Genova Piazza Principe if on the west side of the city) to Pisa Centrale - prices depend on how far in advance you are booking, and on the type of train being booked.