Learn Italian: Il brindisi di Luca mi è piaciuto tantissimo.

The verb piacere (to like) behaves backwards compared to English; it works like the phrase "to be pleasing to". Because of this, it requires essere (to be) as its helping verb in the past tense, rather than avere (to have). You are literally saying "Luca's toast was very pleasing to me."

Why does piaciuto end in an -o?

When a verb uses essere as its helping verb, its past participle (the piaciuto part) has to match the subject in gender and number. The subject doing the pleasing is il brindisi (the toast), which is masculine and singular. If you had liked multiple toasts, it would become plural: mi sono piaciuti.

Could I say Mi è piaciuto tantissimo il brindisi di Luca instead?

Yes! Both word orders are perfectly correct and very common. Putting Il brindisi di Luca at the beginning of the sentence simply puts a little more emphasis on the toast itself, almost like saying "As for Luca's toast, I liked it very much."

How do we get "Luca's toast" from il brindisi di Luca?

Italian doesn't have an apostrophe-"s" for possession. To show ownership, you always have to use the word di (of). You literally have to say "the toast of Luca".