Don’t know about Tavern on the Green; haven’t kept up with all the changes.
Metropolitan is free. They don’t tell you that, but by law they cannot charge you any money since it is sitting right there on Central Park land. They ask for a donation, and it is nice to give something, but I know people who give nothing and walk right in.
MOMA might be free on Friday evenings. You have to wait in line.
Someone mentioned the Carlyle. That would be a good pair with the Metropolitan Museum, since it is nearby.