Buying a bicycle?

You can do it. Lots of bicycle shops. But not all trains will accept bicycles. One concern I would have is the condition of the used bikes. The bikes I have seen were well used even in the rental shops.

You'd only need/benefit from a bicycle in Valencia, the others are all compact enough for walking and twisty/hilly enough to make bikes unwieldy, and the fact is Valencia has great bus service. I've done the same itinerary in reverse, with more time in Valencia and less in Girona.
Whether you go for bikes or not, be sure to get away from the old rambla and the new rambla in Tarragona to check out the updated display of late Roman/early Christian sarcophagi over in the newer section of the town. And leave more time for walking the walls than you imagine, because there are displays worth lingering over.

Thanks for the great advice. Maybe we'll just rent a bike for a day. Speaking of your trip, how did you enjoy the itinerary? Did you have a favorite place that you visited? We saw Tarragona on a day trip out of Barcelona a couple of years ago and loved it so much we decided to come back and see the things we missed on the first go around. We haven't been to any of the other 3 cities yet.

Hi Suze, I agree with others to just rent the bikes for the day of, as they can be quite cumbersome to lug around all the days, you may also be turned away from a bus or train if they are nearing capacity.

Regarding your itinerary, I think a trip down the eastern coast of Spain is a great idea, you'll find the stretch of coastline from Barcelona to Valencia (la Costa Dorada) to be especially interesting. This area has sandy beaches, Templar castles, Roman ruins, posh beach towns, well connected public transport, and relatively less tourists as one would find at the Costa del Sol or the French Rivera.

Some places of interest I would recommend, along your route (north to south):

Empúries - the site of the ancient ruins of a Greek colony from the 6th–3rd century BC. It's quite unique as the ruins sit right on the Costa Brava overlooking the sea. There is a pretty good museum on-site containing artifacts and mosaics of the Roman and Greek periods.

Catalan Medieval Villages - around northern Catalonia there are a number of old stone villages from the medieval times, very picturesque. They include; Besalú, Pals, Castellfollit de la Roca, and Rupit (my favorite). For a nice lunch in Rupit, check out Restaurant Albert for a view of the village below and hardy Catalan mountain cuisine.

Poblet Royal Monastery - a UNESCO World Heritage Site and still active 12th-century Cistercian monastery. It was a fortified medieval royal residence and contains the unique hanging tombs of the old Kings of Aragon.

Les Ferreres Aqueduct - this is an intact Roman Aqueduct just outside of Tarragona, later re-purposed into a medieval bridge. You can actually still walk over it like a bridge. The surrounding forested area is great for some casual hiking too.

Castillo de Miravet - A large riverside stronghold built by the Knights Templar with commanding views of the countryside. The adjacent medieval village spills down the cliff towards the rivers. A great place to bicycle, kayak, or take a leisurely boat down the Ebro river.

Peñíscola - With its cobbled streets and whitewashed houses, the old town of Peñíscola sits on a big rock that juts out into the sea. The town is dominated by a large Knights Templar castle that looks like something out of the Crusades. Recently, Peñíscola was used as a filming location in the T.V. series Game of Thrones.

Albarracín - voted the most beautiful village in Spain, picturesque Moorish fortifications surround this medieval pink-hued village. Was once the capital of it's Berber taifa kingdom, currently in the works of becoming a UNESCO World Heritage Site. A good option for a day trip from Valencia.

Palmeral de Elche - one of the largest historic palm groves in the world, dating back to the Moorish times, designated a UNESCO Site. The Palm forest is surrounded by the town of Elche, one of the oldest in Spain, with much Iberian and Hellenistic archaeology.

Sorry to burst your bubble... but abandoning a bicycle in the street is considered a crime, minor one but a crime nonetheless.

Thank you Enric. Good to know about crimes.

And to Carlos: All of the places you have suggested are much to my liking.

I have been to Las Ferreres Aquaduct but was with a group and had a limited amount of time. We are definitely going back because I want to walk all the way across this time.

Anything dating back to BC is on my list, so Empuries will likely make the cut, too.

Peniscola sounds heavenly as does Albarracin.

Well, they all look great! And please excuse my typos as I cannot put the accents on the place names as should be.

... forgot to add this, in case it interests you while cycling in Catalonia:

http://www.viesverdes.cat/en/
https://www.descobrir.cat/ca/notices/2018/04/6-vies-verdes-per-recorrer-amb-bicicleta-4194.php
http://terra-alta.org/el-recorregut-pas-a-pas/?lang=en

For info: "vies verdes" (roughly pronounced "bee-ahs-ber-dahs") is Catalan for 'greenways', which are sort of "paths for cyclists and hikers" -so to speak- across Catalonia.

And re:leaving your bike behind, I suggest to donate it. There are several organisations that collect second-hand bicycles for those that need them and cannot afford one, be locally, be in projects elsewhere. These are some:

https://www.biciclot.coop/participa/dona-una-bici/
http://bicisparalavida.org/
https://www.bicicletassinfronteras.org/puntos-de-recogida

Hi, glad I could help, and no worries about those accents :)

I'm also a bit of a history buff myself for Antiquity. Not sure when you plan to make your trip, but just last week, the largest intact Ancient Roman mosaics in the world were found in Cuenca Province, Spain, which is a 1 hr on the AVE train from Valencia.

Here was my original forum post on the topic, from last week: https://community.ricksteves.com/travel-forum/spain/travel-news-largest-intact-roman-mosaics-in-the-world-found-in-cuenca-spain

They are still excavating but they plan to have the site open to the public asap, so it may be good to keep an eye out for this.

Oh yes, Carlos, I watched the video and am very excited. Hoping it is open by Oct 1-8. Thank you.

If you're interested in antiquity then while you are in Valencia be sure to get past the obvious art museums, the IVAM and the Fine Arts Museum (which are both great and must-see spots), to go to the Prehistory Museum that is inside the cultural center just another block past the IVAM
http://www.museuprehistoriavalencia.es/web_mupreva/museo_historia/?q=va
This is another case of the amazing resource sitting a stone's throw away from a main attraction but usually empty of tourists -- the IVAM is a draw for the cosmopolitan traveling types [which is why you shouldn't eat anywhere in the area] but they don't seem to make it to the cultural center exhibits very close by. Also an interesting way to see how educational institutions work in Valencia, too.

That usual dining advice (avoid places near museums that draw foreign tourists) does not apply to the restaurant associated with the fine arts museum Museu de Belles Arts, however -- it is an exception to the rule. Excellent versions of local, classic dishes.
Make a reservation early:
https://restauranteelmenjarambviracre.com

La Ruta del Cister. My preference Vallbona de les Monges>Santa Creus>Poblet. A tour would be required without a car. Vallbona de les Monges roads are for road cyclist. Although there is a tour for mountain bikes requiring 4-5 days and hike in 7 days we saw plenty of mountain bikes around Santa Creus. Suspect there are one day bike tours. Empuries is off the public transportation route. For Greek/Roman enthusiasts it is worth a visit. Tarragona sites were closed Monday until June 2. We walked the sites, we were content with views. The Cathedral is lovely. I would suggest two nights in Tarragona.