Tarragona is on the high-speed rail line: from Barcelona Sants the AVE takes 30 minutes (to Camp de Tarragona station, 10 km outside the city); regional trains take 90 minutes and arrive at the much more convenient central station on the seafront. From Madrid the AVE is 2.5 hours.
The nearest international airport is Reus (15 km west, mostly low-cost European flights). Barcelona-El Prat is 90 minutes by road and is the better option for most long-haul arrivals.
By car, Tarragona sits on the AP-7 toll road and the parallel free N-340 between Barcelona and Valencia. Free parking is easy on the western edge of the old town; do not drive into the old town itself.