History and ancient art are always interesting, but want if you want to experience something more modern? Yarat Contemporary Art Space is what's for you. Founded in 2011, it's a non-profit organization - meaning that the entrance is free - with the aim to be a hub for artistic practice, research and thinking. Therefore it's the best place in Baku to enjoy the latest works of contemporary artists without worrying about not understanding: the personnel is here to explain the works and offer curiosities about the artists. You don't need any previous art knowledge to experience what Yarat offers.
Apart from contemporary exhibitions, it oftens hosts retrospectives about modern artists. I can't predict what will you see, but, for example, I had the opportunity to see Love and Protest: the life work of Ashraf Murad, a tormented 20th century Azerbaijani artist who lived to see his beloved country fall under Soviet occupation.
Last but not least, it's in an enviable position, near a park on the end of Baku sea boulevard, a place full of restaurants and bars where you can eat while enjoying the view of the Caspian Sea.