Either is ok. "Should we get something to drink?" has a slightly more positive tone. Just very, very slightly. If I'm expecting (or hoping for) a "yes," I would probably use that. If I'm expecting (or hoping for) a "no," I would probably use "Should we get anything to drink?" But the difference is very, very, very small. VERY small. You can use either. (And either a "yes" or "no" would be polite with either question.)

"Shall we get something to drink?" is correct, and it's considered formal. For informal, you can say "What should we get to drink?" The word anything can be used when you're asking someone if they want a drink: "Do you want anything to drink?"

I think with shall it doesn't fit to say anything I think it's better to say shall we get something to drink not anything

Hello, how's going? I'm a non english speaker so if I'm wrong please let me know! we normally use something in a affirmative sentence when the thing is unknown. When we talk about negative and question sentence we use anything to the same meaning. I hope that helps you