The high season and the best time to go is in May or September, when weather is mild and crowds fewer. Many museums, galleries, castles and the like are only open at this time.
April and October are chillier but you’ll benefit from smaller crowds and cheaper rooms.
In July and August hostels are full of students, especially in Prague.
In winter, you’ll likely get to see it all under a blanket of snow. Attractions in smaller towns are closed, but the time is right for skiing and other winter pursuits.