Depends on what material they’re made of.
Usually for the outsole, a Magic Eraser does the trick. Get it a little wet and then scrub at any stains, and more often than not they come right out. If your toebox is made of a similar material to the outsole, you can usually use the same trick there, too.
As far as keeping the upper white, the obvious tip is to not wear them through mud, grass, or anything else that’s likely to stain them.
For canvas sneakers, like Converse and Vans, a paste of baking soda and vinegar can be scrubbed in with a clean toothbrush, and the stains should lift right out.
For suede shoes, which are SUPER popular right now, a normal pencil eraser can sometimes lift surface marks. Make sure your shoes are completely dry before you do this. A little more care will need to be taken for removing stains that don’t come off with the eraser trick, but there’s a lot of guides online about caring for suede (like I said, super popular right now).
Another technique is gentle laundry detergent in warm water with a soft cloth. This technique is good for most shoe materials.
It really comes down to what your shoes are made of. A quick Google search can yield tons of tips, often broken down by specific fabric. Obviously, the best tip is just to be conscientious when wearing your white shoes, and store them properly to minimize how much dirt they collect when not being worn.
