The Czech Republic has a great system of public transport. You can travel almost anywhere in the country by train or bus. Big cities, larger towns, and tourist locations are interconnected very well with many train or bus connections daily. Many cities have a convenient network of city transport that is reliable and cheap.
Traveling between small towns and villages without a car is a bit harder as there are a lot fewer connections during the day. Your best bet is to take advantage of commute times in the mornings and afternoons when trains and buses run more frequently. If all else fails, you can always call a local taxi company to get you from place A to place B.
Of course, renting a car is always an option if you like to be independent or if you plan to do a lot of traveling around the country.
Taking the Train in the Czech Republic
Taking the Bus in the Czech Republic
Renting a Car in the Czech Republic