Your pottery is safe to be put in the dishwasher, though I recommend hand washing as much as possible as it is much kinder to the glaze surface and less likely to get chipped by other pieces moving around in the dishwasher.

Your pottery has been high-fired (unless otherwise stated) and should therefore be suitable for microwave use. However, my ability to test this is limited and I advise that you exercise caution. Your pottery is not suitable for oven use

As with all ceramics, avoid thermal shock by not exposing your pottery to rapid temperature changes. The risk of thermal shock is greater when pottery is large. Allow freshly boiled water to cool a little before filling your mug. Never put a plate from the fridge into the oven or microwave, never put a hot dish or mug under a running tap of cold water.