It's unlikely that this caused any serious damage. The overflow tank is generally not a pressurized component of the cooling system (on some vehicles it's sealed loosely; on others, not at all). If some coolant splashed out and got on any painted components, it could have damaged the paint, but otherwise it shouldn't do any serious harm. The other possibility is that small amounts of debris got into the coolant in the overflow tank. I would just inspect it to make sure it looks clean and free of debris.
If you do want to flush the system and replace the coolant and you're not 100% comfortable with doing this already, I would have a professional do it (and as usual, make sure you find someone who knows what they're doing). Otherwise you're likely to do more harm than good. On most vehicles it's very hard to get all the air pockets out when refilling the cooling system from low or empty, and if you leave air pockets, you risk overheating and engine damage.