Snap to grid

How do I get Blender to snap to the grid?

When I select snap to grid, it causes any points I edit to move in grid increments … but that’s not what I want. I want them to snap to the grid.

Can I do this without manually entering coordinates for all of my points?

Select the model and press SHIFT+S then “Selection To Grid” first, so at least most vertices will be moved on the real grid when you enable the snap tool

Thank you!