It is very likely for the castles to have moats around the outside. Trebuchets would be valuable to shoot hazardous materials such as fragile containers with boiling oil or water, flaming arrows, human waste, and bodies. Ballistas, siege towers, and ladders would help to breach the tall walls. Battering rams, Archimedes screws, chisels, and wedges would be important tools to drill through the walls. Pontoons or boats would allow crossing the moats. The boats would also be used as a blockade to cutoff incoming supplies and ways for the insiders to escape.
Underground infiltration would be the best to overcome the moat. Going through the plumbing would be way to get inside. People could go through or they could send plague-infected rats through to spread disease. Poisoning the water supply would also prove hazardous to the insiders. Just as the Trojan Horse, we could send in a fake supply caravan, filled with armed soldiers. Mining is also an underground option to get inside the castle. Gundpowder is a handy tool to send through the mines.
Legio I Lynx Fulminata 