Almost every athletic or outdoor shoe brand offers some waterproof models these days. You can look at Merrell, Keen, Oboz, Lowa, Salomon, etc if your taste skews outdoorsy and you want to be able to do some hiking. But Nike, Vans, etc all have some more urban waterproof styles if you’re sticking to sidewalks. The brands Hoka and On are both very popular right now (as much so in Europe as in North America, from what I saw last year), and manage to be both performance-oriented and trendy.
You might want to look for the word Gore-Tex in the product name or description; this is a pretty sure sign that it has a waterproof lining (as opposed to just a water repellant coating). Some retail websites will let you narrow your search by product feature too, so you can look just at waterproof models.
Also, remember that shoes with Gore-Tex or any other waterproof lining won’t breathe as well as a regular sneaker, so it’s a good idea to bring extra socks if your feet get clammy, and go for wool socks instead of cotton or synthetic. I find wool socks are *especially* important for comfort in less ventilated shoes and in rainy climates.
Have a great trip!!