No specific concerns with Omio - third party sites are often very helpful in comparing options and gather information. Some people have reported issues with things such as importing tickets purchased from third party sites into the Trenitalia app making juggling tickets more of an issue. But there are usually work arounds and paper printed versions of purchased tickets work as well.
Most people to use the official site of Trenitalia which is the state train company of Italy: https://www.trenitalia.com/en.html
They also have an app for use in country that also let's you track trains in realtime and validate tickets etc. Use the Italian city names for best results when searching on train sites: Roma for Rome, Venezia Firenze for Florence and get used to 24 hour times for train use.
.Italo is a private train company that runs only fast trains between big cities. https://www.italotreno.com/en
Local train tickets - called regionale - do not come with seat reservations, never sell out and always the same price so buy them when you need them. Tickets in the app need to be purchased a minimum of 5 minutes before the departure time and validated in the app a maximum of 1 minute before the departure time. Paper tickets are available from train machines and these should be validated in the stamping machines as you approach the tracks.
As always great information from the Man in Seat 61: https://www.seat61.com/Italy.htm
Hope that helps, have a great trip!
=Tod