I've checked a map and have another possibility out of Barcelona, but I have no idea about the April and May weather. I suspect April would be quite iffy. I am speaking of the Pyrenees. The hill town of Puigcerda (more or less "poo cher DAH") is about a 2-hour drive from Barcelona. Once in Puigcerda you're in the Cerdanya Valley, which is known for being sunnier than the surrounding areas; it can be quite hot in mid-summer. From Puigcerda the N-260 runs past several nice tiny towns (including Bolvir, Bellver de Cerdanya and Martinet) to the very handsome larger town of La Seu de Urgell with its historic cathedral. La Seu looked liked it would be a nice place to spend the night, as is Puigcerda.
In the opposite direction you can cross a few miles of France to reach the Spanish (!) town of Llivia, which is larger than Bolvir, BdeC and Martinet and had multiple restaurants open when I visited the area in July. All of the above could be covered by public bus on a summer weekday, though not in one day. I assume the buses run all year long, because they serve primarily locals.
You're not yet actually up in the mountains, but they are visible from the previously mentioned towns. For actual mountain time the French Yellow Train is a treat. Much of its route is through the mountains. The train runs from La Tour de Carol to Villefranche-de-Conflent, and the entire trip takes between 3 and 3-1/2 hours one way. Since I was staying in Puigcerda, I opted to get on at Bourg Madame (about 2 miles from Puigcerda). There's parking at the Bourg Madame train station; I assume it's free.
Some of the trains run only part of the route, between Font Romeu-Odeillo and VdC, and there are more trains during the summer (defined as beginning on June 25 this year), so verification of the schedule is a must. The online schedule doesn't extend to April and May yet.
Many riders did only part of the trip, but I'm betting the access road(s) along the train route have some scary moments.
VdC is an extremely picturesque fortified UNESCO-recognized town with a walkable wall. It is very touristy. One of the other stops is near Mont Louis, also fortified and a UNESCO site. It's a bit of a walk from the train station serving Mont Louis to the town; an electronic map is advised, in case there are no people around to point you in the right direction. I was switching to a bus at Mont Louis and unfortunately didn't have time to go inside the city walls, but there certainly were not many tourists visible outside the walls, and I'm confident that it is less overrun than VdC.
Yellow Train tickets can be bought from SNCF. I did not need to buy a ticket in advance. This is a local train, and I doubt that a seat reservation is possible. Neither train I rode was full, though there were a lot of people making the northern part of the trip.
For checking the schedule, use Villefranche Vern (Villefranche Vernet les Bains) as the destination. There were schedule leaflets available at the Bourg Madame station this summer. Bourg Madame has a tourist office not too far from the train station. It's easier to find the T.O. than the station, which is down a side street near a butcher shop. I don't remember whether there is a sign marking the turn.
