Taormina's night is the passeggiata along Corso Umberto, drinks on a terrace with a view you pay for, and one long-running late bar; from January to March a good deal of it is shut.
Morgana Lounge Bar
The town's late bar, in a courtyard off Corso Umberto, open daily 8pm to 3am according to its own site. Its season runs April to December and it is closed outright from January to March, which tells you most of what you need to know about Taormina in winter.
Corso Umberto passeggiata
The main street is pedestrianised and the entire town walks it between roughly 7pm and 11pm, in both directions, several times. It costs nothing, it is the actual social event of the evening, and the bars along it fill from the passing traffic.
Wunderbar Caffe, Piazza IX Aprile
The terrace cafe on the belvedere square halfway along the Corso, with tables facing out over the bay. You are paying for the position rather than the drinks, and it is the standard place for a first drink before dinner.
Mazzaro and the beach clubs
In summer the lidos in the bay below town serve into the evening, which is a different night from the Corso: sea level, less dressed up, and reachable by the cable car from Via Pirandello. Check when the last funivia runs back up, or you are paying for a taxi.
Off-season closures
Taormina is a resort town of around eleven thousand people and it empties in winter. Between roughly November and March the beach clubs shut, the cable car runs a reduced timetable and many bars and hotels close for the season, Morgana included.