I have done the journey from reus to Vinaros many times via public transport. Tortosa is easier.
Here are the options
There is a HIFE coach every day from Reus at about 5pm.
(www.hife.es)
There is a local Reus - airport bus that drops off at the bus station and the railway station. (www.reustransport.com)
There is a direct Tarragona bus from the airport that terminates at the bus station (Hispano Igualadina) which has an infrequent timetable. 15 min walk to train station.
There is also a bus service to Salou serving most of the hotels and goes near to the station
Cheap trains from Tarragona every 90 mins to Tortosa