I agree with the posters who suggested that you buy a used bike and just use it while there. You would not have any trouble finding one anywhere in Spain --- I live in a small town and there are ~5 bike shops selling used bikes there.
This being said, OP said he wants to be able to take his bike while travelling in Europe. For that, yes a folding bike is a good choice and on internet you can find instructions on how to fit them in large luggages. Watch out for the weight though.
In Spain, you can easily find cheap, low-end folding bikes. Brompton clones and such go for few hundred Euros. Larger, more bulky ones for ~100 Euros.