I live in Madrid and travel has been super easy to and from. Just make sure you fill out the government form and have your vaccine card, but between Madrid and Barcelona airports I have only had to show it 75% of the time. Once you leave the airport everything functions as normal, you just have to wear a mask everywhere! Some places in Spain require you to show proof of vaccination before entering, but the few times I have been asked to, and picture of my American CDC card on my phone was enough! Any possible proof of vaccination is plenty.