1

Beginning of May is still too early for higher mountains, but the southern Spanish side, and a bit lower altitudes, should be fine. Good trail conditions and good weather are hit or miss, there are years with little snow and years with tons. You can have a +30C heat wave and +10C and rain. Whether or not you'll enjoy it depends on what you expect, as in, you shouldn't expect summer.

Public transport connections are crappy, a hiking base with several day hikes easily accessible by public transport or close to an airport - dream on!

2

The Pyrenees are fairly close to Barcelona airport. Not on it's doorstep, but about 4 hours by 'Alsa' bus.

3

Why dont you try the are of Sierra Nevada NP in southern Spain?
It has our national summit (Mulhacén 3 480m), the airport of Granada is very close and the weather in May is vey nice.

4

The Pyrenees covers a distance of 866 km (538 mi). Which part are you thinking of?

5

My knees don't allow much hiking but I can admire the mountains from lower altitudes (or roads).
My favorite area is the south of France from the Med to the west...Languedoc-Rousillon (Aude and Pyrenees-Orientales)...Ariege---

First of all, the Pyrenees-Orientales are both lowlands and mountains, the main city if Perpignan which is north from Barcelona (or Girona) and an airport in Rivesaltes/Perpignan also. with good railroad service. The train service is limited...take a SNCF train to catch Le Train Jaune...which will take you all the way to the top--but whether it's too much snow or not, can't be sure.
Have you checked out the almost omniscient Man in Seat 61
http://www.seat61.com
He'll give you all sorts of help on finding routes.
Here's some info on the PO:
http://www.epyrenees.com/guide/activity.php?activity=6&region=E
http://www.lespyrenees.net/en/randonnee,2,Information-on-Hiking-Trails.html
http://www.tripadvisor.co.uk/ShowTopic-g1774988-i17937-k5533204-A_few_hikes_in_Pyrenees_Orientales-Pyrenees_Orientales_Languedoc_Roussillon.html
http://en.argeles-sur-mer.com/articles-tourisme-3/41-61-walks-hikes-pyrenees-orientales/
A writer in the Guardian spoke of her experience having taken the Train Jaune...and gives a link to:
http://www.tourisme-pyreneesorientales.com/en/449/pages/d/0/0/page/0
info on the Train Jaune
http://www.tourisme-pyreneesorientales.com/en/4791/pages/d/0/0/page/0
Also go to the Press Section (top of page, on left) and select Hiking for a pdf download
(and check out others).

For hiking in the Ariege, you could fly into Toulouse and take the train to Foix and further on into the Pyrenees:
Foix to: Ax-les-Thermes, L'Hospitalet-pres-l'Andorre
Bourg-Madame, Mont-Louis, Pulgcerda, Villefrance-de-Confluent (which is where one finds the Train Jaune when coming from Perpignan/Rivesaltes.
http://www.ariege.com/what-to-do-in-ariege/walking-hiking/index.html is a good intro...

My favorite guides for this area are both Rough Guides: Languedoc-Rousillon and The Pyrenees. Not a lot of fancy pictures but reliable info from people who wrote the book and come back to revise it every couple years.

But those are just a couple sections of the chain...just my opinions, biased ones at that..
whatever you choose, hope you have a great time

6

You have Parc Natural del Cap de Creus in the north of Catalonia, Zona Volcànica de la Garrotxa as well or Parc Nacional d'Aigüestortes. All of these are reachable from Barcelona.