After the scenes in the Floating Ruins and aboard the Celsius, speak with Brother - the guy on the floor - and opt to comfort him. (Also feel free to hear him groan as you walk on him. Repeatedly.) Also speak with Shinra to obtain the Black Mage Dressphere - this can be critical in hitting enemies’ elemental weaknesses, which is generally a good thing.
Other than that? You have some missions to do in Besaid, Kilika, and Zanarkand - you can see Chapter 1 - Besaid, Chapter 1 - Kilika (a brief trip), and Chapter 1 - Zanarkand if you want to get directly into the meat of the game. However, we highly recommend going everywhere else (with the exception of Kilika) first, especially if you intend to go for 100%. In any case, here we go.