You might practice with an older pair of shoes to get the hang of it before you do your new ones.
Take them to the sink with an old toothbrush or fingernail brush. Start with a soft bristle brush, go to a hard bristle later if you need to.
Use any liquid soap. With your sink sprayer get the shoe wet (we are just talking outside, not inside.). Put the liquid soap on the shoe and rub with the brush. You might start with just a cloth on the shoe but use the brush on the sole and band between the shoe and the sole.
Scrub a bit, rinse, scrub, rinse. The soap will continue to loosen the dirt and you rinse it away. Don't scrub hard at all on the shoe, it can go to deep into the shoe. I wear white reeboks.