Amsterdam, Paris, Barcelona

Where does one start... You need to do this research yourself as there is so much to see & do. Not knowing what you like, how can anyone advise. Start with RS books which you should be able to get from your library if you don't want to buy them right away. Once you determine what interests you, ask specific questions if you're interested in someone else's opinion.

?
No one can advise you. Only you know what will
interest you. Do a google search for places of interest in each city you visit.

Try 'Amsterdam sights' or 'attractions'. Do the same
for Paris and Barcelona. Read about each and make a
list of those places which interest you most.

You will never have enough time to see them all; so
put a priority on your list.

Abraham, your question of "what sites or activities are worth doing" is why tour books are written. The guide has spent the time to explore the area(s) and list the options; a good book, like the ones RS writes, offer opinions about what to do and what to skip or make a lower priority.