A Coruña

In the Cidade Vella, you can find among others:

Castelo de San Antón
This Castle holds an interesting Archaeological museum. It used to be located in an island.
Church of Santiago
Located in the Old Town, the church is originally medieval, but has seen major changes.
Colexiata de Santa María
Now hosting the Museum of Sacred Art
Museo Militar
The Military Museum holds collections of weapons, uniforms, maps and so on.
Museo Emilia Pardo Bazán
A museum dedicated to one great Spanish writer of the XIX century. The building also houses the Galician Language Academy
Xardín de San Carlos
A beautiful and romantic garden. Often used for wedding photoshoots, and where the remain of Sir John Moore a British General that fought in the Elviña battle during the Napoleonic wars lie.

Some other sights:

Torre de Hércules
Roman lighthouse. Apparently, this is the oldest lighthouse in the world that is still in use.
Castro de Elviña
Castros are Roman or pre-Roman settlements. This particular castro has been neglected for a long time, and is undergoing major archaeological work. Many findings are on display on the Castelo de San Antón.
The port
Casa de las Ciencias
Science Museum
Domus
Another science museum, dealing with the human body.
Aquarium Finisterrae
An aquarium, very close to the Domus.
Picasso's home
which you cannot visit; there is simply a small plaque to identify it.
María Pita
´s square, with the Council Building
Fine Arts Museum