Whatever happened to solid bike tubes?
You should not be getting multiple flat tires in such a short timespan. I commute on poorly-maintained roads in Atlanta, and have not had a flat tire in 4,000 miles.

In order of estimated likelihood, either:


your tires are not properly inflated
your tires are worn or punctured and need to be replaced
you have a sharp object embedded on the inside of your tire
you are exceedingly unlucky


If you've just started cycling again, I think it's exceedingly probable that you've simply neglected to inflate your tires to their recommended pressure. Standard road tires should be inflated to pressures of 100PSI and beyond. Mountain bike tires (which are awful for road use, but I digress...) require much less. It's easy to grossly underestimate how much to inflate a bike tire, and a standard hand pump will likely only get you to 30-60 PSI before you give up.

Take your bicycle to your nearest bike shop and have them examine the situation. Flat tires should not be a common occurrence.