What could be causing red and blue/purple-ish splotches on my legs after shower?

Warm water. Warm water can cause skin blood vessels to dilate and bring more blood close to the skin surface in attempt to cool off. Try using cooler shower water.
After shower. Dry or sensitive skin may develop such splotches after a shower, but if not intensely itchy it is not too problematic. If you note dry or sensitive skin, apply a moisturizing agent like alpha keri lotion while your skin is still wet to retain some skin oil and moisture.