Learn Spanish (Latin America): ¿Quieres un refresco o una cerveza?

The verb quieres already means "you want" because of its -es ending. In Spanish, it is very common to drop the subject pronoun unless you need to clarify who you are talking about or want to add emphasis.

Does refresco mean any refreshing drink?

While it comes from the idea of refreshment, in most of Latin America un refresco specifically refers to a carbonated soda or soft drink rather than just any cold beverage.

If I wanted to reply "I want it," how would I use object pronouns here?

You need to match the direct object pronoun to the gender of the drink. For the masculine un refresco, you would answer Lo quiero. For the feminine una cerveza, you would answer La quiero. If someone offered to serve it to you, you would see the double pronouns from this current lesson: Te lo sirvo (I serve it to you) or Te la sirvo.