What path does a Sun-like star follow after exhausting its core hydrogen?

It swings sharply up and to the right into the Red Giant branch.

For a star of low to intermediate mass, such as our Sun, the evolutionary change after exhausting core hydrogen is dramatic and clearly mapped on the diagram. The star begins to expand significantly while its surface cools, causing its representation on the plot to move sharply upwards and towards the right, entering the Red Giant branch. This movement signifies a temporary phase where luminosity increases greatly while temperature drops. Following this expansion, the star may eventually shed its outer layers, leaving behind a cooling White Dwarf, which then traces a path almost straight down the diagram as it gradually loses heat.