A lot of good advice from the above posters (except the recumbent advice – JK Guys!).
There is a good chance that you pain may be from bad form. Many times riders ride with what I refer to as “rigor mortis” where they lock out their elbows & grip the bars so tight; it causes engery sapping muscle tension & ultimately cramping, stiffess & soreness felt in the hands, arms or the shoulders.
As you ride around, make sure your shoulders are relaxed, elbows are bent & hands are holding the bars like you would with a small child across the street – firm enough not to let go but gentle enough not to crush their hands.
Also, as you ride; make sure you change your position. One way to do this, is start developing different habits for how you ride. For example, when I hit a hill or am resting; I use one hand position on my bars; for steady riding; I use another position & when I sprint or ride hard; I may use another. Practice riding in the drops, the hoods & the top bar – if you rotate those positions through your ride; it will help break the lactic acid up & make you more comfortable.