As for the public transport, this differs city by city. In Prague, most of the ticket machines only accept coins, but the major hubs (e.g. at airport, major train stations) are recently being outfitted with ticket machines that also accept payment cards (and banknotes). However I noticed some of them only accept contact-less cards.
Major train stations all accept payment cards.
Virtually all supermarkets and most of shops accept payment cards in larger cities. Even in smaller ones, supermarkets will often accept payment cards.
As for types of cards accepted - almost all domestic payment cards are issued by Visa or MasterCard, so you won't have problems with those. Don't know about others. The accepted authorization methods are contact-less, chip-and-pin, often magnetic stripe and pin should be possible. In the past, magnetic stripe and pin or signature were used, so many terminals still have the stripe reader. But I wouldn't rely on signature being accepted instead of pin anymore.