The easiest way would be to take a taxi right at the airport. But it will be also the most expensive.
BCN airport is quite bad connected with train, and the trains arriving at the airport link with Barcelona, not with other cities. On the other hand Barcelona is well connected with Tarragona by rail; maybe you looked for "Camp de Tarragona" station? This is the High Speed station and is in the middle of nowhere, and few trains stop there. A quick search for today shows many options (but discard any train stopping at Camp de Tarragona, "camp" means "field" in catalan, farmers land)
I think a easy connection yet affordable would be to take a taxi at the airport to Barcelona Sants station and then take a train to Tarragona. An even cheaper solution would be to board the local 46 bus at the airport terminal, drop off at Plaça Espanya, then take the Metro green line (L3) and get off at Sants-Estació to take the train. As Gagravarr notes in his comment, the Aerobus express service is also a well balanced option between price and convenience. This express bus has room for luggage, for instance, while the local bus doesn't. It all depends on your particular needs.