Don't worry in this situation. Whatever you say, they'll figure it out. Since they make money when you stay there, simply by being their customer you are automatically making them happy.
I walk through the doors. I'm in the lobby. I see the "desk" (usually a big counter) ahead of me. If there's no line I walk up to it. If there's a line, I get in line.
What I say may depend on whether I have a reservation. If I have a reservation I know they will need my name so I give it to them right away. Often the clerk speaks to me before I have a chance to speak to them. Here are some ways the conversation might go?
Clerk: "Checking in?"
Me: "Yes, Dan Smith, checking in, please, with a reservation."
Me: "Checking in."
Clerk: "Oh, hello, sir. Can I help you?"
Me: "Yes, please. Dan Smith, checking in, with a reservation."
Without a reservation, at a big chain hotel or motel on a highway--neither cheap nor expensive:
Me: "Hi, I need a room for two nights for two people."
Clerk: "Yes, we have one. Two twin beds or one king?"
Me: "One king, please."
Clerk: "Smoking or non-smoking?"
Me: "Non-smoking, please."
Clerk: "We have one for $119 a night."
Me: "Do you have anything cheaper?"
Clerk: "Yes, we have one on the second floor for $109."
Me: "That's fine. Do you have a discount for AAA?"
Clerk: "Yes, we have a 10% discount for AAA."
One travel tip: in the United States, there are many small add-ons--various kinds of taxes and fees and such--that add up, so that "$109" room may end up costing $123.45. If you are on a tight budget, once they actually have you and the room in their computer, it is very reasonable to ask
Me: "With the taxes and everything, what will get charged to my credit card?"
Clerk: "If there are no additional charges, that will be $123.45."