Congrats on your first eggs!
You are going to want to remove any eggs. Check once or twice a day to remove them, more if they start eating the eggs.
A fair warning: chickens do eat eggs. Not all do, but try to avoid letting them have the opportunity.
Try to avoid cleaning if possible. There's a protective membrane around the outside that helps keep the eggs fresh longer. It will be taken off if you clean them. If the eggs are clean, don't wash them and you can store them on your counter at room temp for up to two weeks or in the fridge for up to a few months.
Clean them before use of course.
If they are especially dirty wash them under cold water with a brush and immediately dry with a paper towel or clean cloth. Store clean eggs only in your fridge for up to about two months.
Hope this helps!