How can I build a dome on top of a building, similar to a state capital or something similar?
The building has a base of 16x24 studs.
Like oezi said you can build a dome out of cascading bricks, which is perfect for most scales of building. For smaller scale building where you'd want a less jagged approach you may want to use a combination of 1/4 dome pieces. A complete dome will take up an area of 6x6 studs wide.
This technique is common in the Prince of Persia sets.
For a true miniland scale building you may want to have a closer look at the model in Legoland Florida.