Hi Marlin, welcome to the forum.
The SOP is to remove the grout to a depth of 2/3rds the tile, then grout it. Is this your typical cement based grout...or an epoxy grout?
The best grout to use depends on what your application here is and what kind of tile you have.
Yes, a grout float is the tool to work the grout into the groutlines (and the pits of the tile if that's what needs to be done). We can't "see" what you have there unless you share. Can you tell us if this is a man made ceramic/porcelain tile...or a natural stone tile? How wide are the grout lines? What room is this floor tile in? Perhaps you could post a picture?