We shall stay in Madrid not far from Puerta de Alcala.
Are there any taxi stands in that area, please?
Is it easy to catch a taxi in the street, just waving the hand?
Yes, it's easy to just wave them down. I didn't do it but saw people doing it, and the employee at my hotel in Madrid went outside and waved one down. Another option is to use the PIDETAXI app which orders you a taxi and it's all in English. Here's my recent trip report, day 11 and onwards is about Madrid and taxis. https://community.ricksteves.com/travel-forum/trip-reports/trip-report-spain-november-2023
Madrid’s center is very walkable and unless a manner of your party has mobility limitations there should be much (any) need for a taxi during your visit.