Toulouse to Tarragona

By train.

Toulouse → Narbonne, then Narbonne → Tarragona

Trains in France:

https://www.sncf-connect.com/en-en/

Trains in Spain:

https://www.renfe.com/es/en

When is this going to be? Sometime this summer RENFE will start a direct Toulouse - Barcelona service.

Till then just book Toulouse - Barcelona with SNCF.

What is best to get to Taragona from Barcelona also depends on where in Taragona. There are high speed trains, but they stop outside town.
There are regional trains from Barcelona to Tarragona at least once per hour, and tickets for those you can just buy once you are there.

Even better Renfe has their once a day train that will take you from Narbonne to Tarragona without having to change in Barcelona. It leaves Narbonne some time after 10am. I just took this train roundtrip Montpellier, France to Madrid.

There is indeed that train from Narbonne to Tarragona, but calls at Camp Taragon, which is outside of Tarragona. It also means that you change from a SNCF train to a RENFE train in Narbonne. If the SNCF train is late you will be stuck there. Which is why I think it is better to travel on SNCF all the way to Barcelona, and then take a local from there.