Yes — each book in the trilogy should be read in order. Legendary reveals Tella's perspective on events from Book 1, and Finale continues directly from Book 2.
The Eventyr series (Once Upon a Broken Heart, etc.) is set in a different world from Caraval. Some Caraval characters appear briefly in the Eventyr books. Read either series first — they are largely independent.
Caraval is most often compared to The Night Circus by Erin Morgenstern (atmospheric magical performance setting) and Six of Crows by Leigh Bardugo (competitive stakes, morally complex world). Readers who loved either often enjoy Caraval.