The best shoes for rainy weather use materials that don't easily absorb water. The worst shoes are made from highly absorbent materials like suede or mesh. These soak up water quickly and hold onto it, drenching your feet, even if it's just drizzling. And even if you don't mind the wetness for a little while, materials like suede are sensitive to water and can develop water stains if they aren't cleaned immediately and properly after getting wet.
When shopping for weatherized shoes, it's important to understand some of the terminology. "Waterproof shoes" and "water-resistant shoes" aren't the same. The former means no water gets into your shoes, while the latter means the shoes keep water out up to a point, but not completely. Some leathers and synthetics are better than others because they're treated with a protective coating that wicks water away better, so look for these weatherized versions.
For example, shoes made with water-resistant materials like leather and synthetic textiles might keep your feet dry in a light sprinkle, but if you step in a puddle, your socks will be soaked. Premium waterproof materials like GORE-TEX are different. This material is 100% waterproof, with teeny-tiny pores too small for water droplets to seep through.
Water-resistant and waterproof shoes are much easier to clean after a downpour or walk through puddles. They're less likely to stain, and you can usually wipe the exterior clean with a towel.