I use a fairly firm grout application float in a rotary movement - or is that a sweeping movement more - remembering to change direction so that the grout gets fully into the gaps. Once the grout starts to dry (when it starts to powder on the
tile face) I start to use the grout cleaning sponge. I use a backed sponge as I find it is far, far better than the "traditional" hand-held sponge still used by many - particularly as it doesn't dig into the
tile gap unevenly, removing more grout that you would intend. Currently, I am using a Rubi one, with a wood handle - excellent. For grout cleaning-off you need a sponge that is cut into quite small squares for the best result. Of course, I have had a lot of advice from
Italian tilers on this as every house here has tiles everywhere!
Get hold of a wash bath with a couple of rollers so that you can keep your sponge clean and not too wet whilst you are working. The better tilers here will tell you to sponge over three times, changing water whenever it gets too dirty.
Of course, it depends on your grout (Mapei Ultracolor Plus is excellent) and the quality of your sponges, but the job is fairly straight-forward once you get both the hang of it and the right tools for the job as well.
This is a Plasplugs Grout Wash Tank - a bit DIY, but the only one I could (quickly) find for you to get the idea - as well as my not knowing what kit is on sale in the
UK anymore