Learn Italian: Preferisci la doccia o la vasca? - Do you prefer the shower or the bathtub?

Preferire (to prefer) is one of the many -ire verbs that add -isc- to the stem in the present tense for all singular forms and the "they" form (io preferisco, tu preferisci, lui/lei preferisce, loro preferiscono). The "we" and "you all" forms don't use it (noi preferiamo, voi preferite).

Is vasca the full word for "bathtub"?

The full term is vasca da bagno (literally "bath tub" or "bath basin"), but Italians usually shorten it to just vasca in everyday conversation, similar to how English speakers just say "the tub".

Why do we need the article "la" before both options? Can I just say "Preferisci doccia o vasca?"

In Italian, you almost always need an article before a noun, even when giving a general choice. While in English you might sometimes drop the article and ask "Do you prefer shower or bath?", Italian strongly prefers keeping the definite articles (la doccia and la vasca) for both items.