Even if you're riding in warm, dry conditions, your cycling shoes are likely to get sweaty inside. Once you take them off, don't just leave them sitting in a plastic bag, but put them somewhere dry where they can air and dry out. It's a good idea to pull out your insoles too, to make sure that the inside of the sole unit dries out. Many more expensive cycling shoes will come with a net storage bag to keep them in.
If your shoes get wet, you should take out the insoles and pack the inside of the shoes with scrunched up newspaper to help soak up the moisture.
Most cycling shoes can be wiped down with a damp cloth, but don't try anything more aggressive to clean them unless the maker's care instructions advise this.
If you're regularly riding in poor conditions, a pair of the best cycling overshoes will help keep your shoes clean. Overshoes tend to wear quickly though; a dedicated pair of winter cycling shoes like the Gaerne Icestorm boots might be a better bet. They will have a fully enclosed, waterproof upper and sole, usually with a breathable membrane and insulation, to help keep your feet dry and warm.
Heel support and heel lift are often an issue with this type of winter boot though, so make sure that the fit is good if you go this route. Also consider swapping to winter boots and pedals for two bolt cleats for their better walkability in the wet or when it's icy - you're less likely to take a spill on a tile floor at the coffee stop.