Answer: To be very basic you could build a set of one of each denomination of U.S. gold coin produced for general circulation, the $1, $2.50, $3, $5, $10, and $20. Beyond that, a popular set to form comprises the following late 19th and early 20th-century issues: gold $1, Liberty Head $2.50, Indian $2.50, $3, Liberty Head $5, Indian $5, Liberty Head $10, Indian $10, Liberty Head $20, and Saint-Gaudens $20.
If you want to have a more extensive set, still staying within the same date range, your set could include three basic main designs of gold $1 pieces, the Liberty Head $2.50, the Indian $2.50, the $3, the Liberty Head $5 No Motto, the Liberty Head $5 with IN GOD WE TRUST, the Liberty Head $10 No Motto, the Liberty Head $10 With Motto, the Indian $10 No Motto, the Indian With Motto, the three types of Liberty Head $20, the MCMVII $20 (if you are able to get into this price range — it certainly is a beautiful coin), the Saint-Gaudens $20 No Motto, and the Saint-Gaudens $20 With Motto. This set is a very beautiful display of American gold coinage.
Of course, you can later expand any of these sets and go into the elusive issues produced 1795 through 1834, all of which are rare.