My corrections are in red and comments in blue.

My last holiday

Last time I went around on a trip it was a few months ago.
During Easter I spent a week in Madrid with my boyfriend. It was an short holiday short but sweet!
We visited the biggest part of the city by walk on foot and we moved by took the underground (in North America we call it the subway) in the only for longer distances only.
I think that if you want to really know a place and you want to take beautiful photos, you will should walk along it. You have more possibility possibilities to discover magical places not showed shown in the brochure: little squares unless without tourists, narrow historical streets, typical good (not crowded!) food (I rearranged this sentace) local restaurants with good food (and not crowded!), and many other things you can found find step by step. In That way every vacation could become an amazing surprise!
At the same time, I am extremely organized so I set up what will do for each day. I and my fiancé are keen on architecture, especially contemporary (then we love walking around,not sure what you are trying to say here) but we also visited also the Historical Garden, the Crystal Palace, the Royal Palace and many ancient churches. We spent a whole day at Del Prado’s Museum, where we saw modern paintings and sculptures and I finally could watch at saw one of my favorite paintings: the Guernica of Pablo Picasso. It fitted('It took up a whole wall' is better) a whole wall! It was enormous!
I remember I stayed locked in front of that painting for fifteen minutes, it showed the tension and the suffering of people during a German air raid. It was really impressing!
Absolutely casually I got together with an old friend of mine: Sarah. I haven’t even seen her since I was eleven! What a strange(strange is an adjective, must have a subject, for example: it was strange, what a strange surprise)! She told me she was a waitress in an Italian restaurant. We had a funny happy hour together talking about our past life.
The last day my alarm clock didn’t work(we say 'didn't ring' in this case) and we woke up only two hours only before our flight takes off. We run ran very fast! It There was an accident in the subway, so we got in a taxi and we arrived at the airport in time but it was too expensive!
I told me myself (reflexive) next time I will set up two alarm clock at least!

Well written, lots of character in your descriptions! Keep up the good work.