Learn Italian: Vorrei prenotare la sauna e un massaggio rilassante. - I would like to book the sauna and a relaxing massage.

Vorrei means I would like. It is the conditional form of the verb volere (to want). In Italian, just like in English, using the conditional makes a request much more polite than saying voglio (I want), which can sound demanding when asking for a service.

Why is the verb prenotare left in its dictionary form?

Because it immediately follows vorrei. When you use forms of volere to express what you want to do, the main action verb that comes next always stays in its infinitive (dictionary) form.

Since massaggio is a masculine word, why does the adjective rilassante end in -e?

Not all adjectives end in -o or -a. Rilassante belongs to a common group of adjectives that end in -e in their base singular form. These adjectives use the exact same ending for both masculine and feminine nouns.

How would I change this sentence to ask for two relaxing massages?

You would ask for due massaggi rilassanti. The noun massaggio drops its final -o for an -i, and the adjective rilassante changes its -e to an -i to become plural.