I don't know if this will work for you, but it did for me. When I first began dancing I had flat feet (a trait I believe I inherited from my father). Anyway the instructor told me that before I go to bed every night I should place a pencil at the heel of my foot and roll it until it to my toes then I should try to pick up the pencil with my toes. She said that once I was able to pick it up I should hold it in a point for a few minutes. I don't know how but it eventually worked and I did develop an arch in both feet from that exercise.
With flat feet you need good shoes with good arch support. Buying your shoes at a bargin shoe store or department store is not a good idea. Go to a shoe store that has a certified podorthist and be fitted for a good pair of shoes.
A flat foot in most cases is a normal variation of the shape of the foot. It is often hereditary. You don't have to do anything to them as long as they are not hurting you. If there is pain, it is usually associated with another condition that may need treatment. An othotic (arch support) in your shoe may help.
I developed very painful foot pain that i had to wear a "walker" as my foot doctor called it it's a boot that you strap on. That was the only thing that helped me walk. Finally doctor precribed an insole with arch support specifically for my foot and it's really helped. Maybe you should have a doctor precribed you also. But find out first if your insurance covers it cause it could be expensive. Mine is worth $400. Good Luck. Let me know how it goes.
Basically you should go to a podiatrist for a full assessment. The net result will probably be orthotics (custom made arch supports) and whilst expensive they are worth it long term. Some people dont agree with giving orthotics if you are pain free, you will get different opinions on this and should decide how you feel - in many cases i feel they are warranted to reduce the rate at which you get things like osteoarthritis, heel pain and arch pain later on. Footwear is aREALLY important part of all of this, but i would reccommend you look into the orthotics before you go shoe shopping and then take your new supports with you when you go shoe shopping. Once you have orthotics the minor arch supports sometimes already in a shoe can destabilize the support, so go to a good shoe shop that claims to and apears confident in fitting shoes that will take your orthotics. Good Luck. PS You will also get varying opinions, but i dont personally believe foot exercises will do much long term - it may strengthen the muscles/tendons and support structures slightly, but will not change the arrangement of your bones!!