What are my rights if the bartender refused to serve me at a bar?

Any owner or manager of a private establishment can refuse service to anyone for any reason as long as the reason doesn't deal with a protected status such as race, age, ethnicity, gender, etc. In addition, a drinking establishment and/or the drinks server can be held liable for the criminal actions of a patron who leaves the bar dangerously drunk and then hits someone while driving drunk.

A bar is not required to serve you. In fact there are many restrictions on who they can lawfully serve. For example, you must be over 21. Also they cannot serve someone who is visibly intoxicated. This is called Dram Shop Act liability. But even if neither of these was the case they can refuse service to anyone they wish so long as it is a non-discriminatory reason. Your rights are to get up and go to another bar.

It depends what the reason was for not serving you. Alcohol servers are required to refuse service to anyone they think may be already intoxicated, including by any drug. It's a tough judgement call that alcohol servers are trained to make and required to make. They are required to err on the side of safety. It's not easy. If you look, move, or talk like you are intoxicated, they can't legally serve you. In a place where everyone is drinking, servers are required to pick out, merely by observation, which patrons look as if they pose a risk. Many states have dram shop laws that hold the bar responsible for the actions of a patron who has been served alcohol when they should not have been. Many states also require training courses before a person can work as a bartender or alcohol server. They also cannot serve you if they think you are underage. Sometimes a server will not card a person who looks too young, because they do not want to have to ask the person to leave, but still will not serve the person a drink.