There exist special little tile grout knives, which are really just a blade with carbide dust on the edge. They are efficient at cutting out old grout between tiles.
There is a special a power grout remover as one of the accessory blades that comes with the FEIN MultiMaster. Unlike a circular saw blade, this partially round carbide dusted blade oscillates back and fourth, and removes tile grout very well, thank you. The flat side allows you to work right up to a wall, where a round blade hit the wall before finishing the groove.
Another blade on the same tool is simply an oscillating knife, that does a great job of cutting glue off of the substrate.
Follow this link for a video on removing grout and this one for a video on removing glue.