If you are not travelling on a budget you can simply go to the station, pay the top walk-up fare for the first high speed train departing and get on.
Unfortunately flexibility is not free. On trenitalia.com/tcom-en the walk-up fare is called BASE. On italotreno.it/en it's FLEX.
Base and Flexi fares are displayed in advance together with the discounted ones. Discounted fares can't be purchased on the day of travel and come with restrictions (no changes, no refunds). Go on the above sites, enter you days of travel and you'll see how much money you could save.
Italotreno is a private competitor and it's often cheaper than Trenitalia. Follow them on twitter, or subscribe to the company newsletter, to receive promo codes.
Except for long week-ends and early/late august, high speed trains do not sell out. The slow local trains can't sell out.