We went to this museum last week. For an amount of AED. 3 (adult), you'll get to know so much about Dubai. The place is well-organized and fascinating. I love that it is sectioned to what the community of Dubai can offer (Land, Sea, etc.). It showcases the history as well as the real culture of the place. I never thought educational trip can be so much fun and the staff are friendly and helpful too (especially the guard stationed inside who even volunteered to take a picture of me and my husband, thanks!).
To go there, take bus number C7 if you go down to Al Ghubaiba Train Station.
And as a bonus, you can go afterwards to Abra-Souk area, well, that's another fun part touring Dubai.