Wall tile grout can be cleaned using the likes of oxalic acid. This acid should be tested in a small area prior to applying over the tiled surface using the recommended concentration and appropriate PPE. I would not recommend using this with natural stone as the surface will not be so resilient and may discolour.
It depends how bad it is, a good bleach containing abrasive cleaner like Jif and plenty of elbow grease with a quality plastic scourer is the lower end of the scale, or de/ regrouting; either way its a bit painful and labourious. Not quick fix I'm afraid.
You can buy pens that will go over the grout lines, but if your grout is discolouring it may be that the wrong grout has been used, in my opinion it is better to remove & replace with antifungal grout as this stops this happening.
The best an cheapes solution is to use ( not chemicals provided for tiles which will cost a bit more) is thick bleach the cheapes from any market .
It smells chlorine but do a job and prevent for quite long time from mould groving and yellow soap stains.