You don't mention your fitness level, or how hard you are riding.
Riding 10-15 miles (16-24 km) straight away sounds too much for you.
I recommend you aim for 5 miles, every second day. While riding, consciously change your hand position between the tops (the straight part of the handlebars), the hoods, and the corners of the bars, with an occasional stretch down to the drops. Stop a little after you feel those muscles getting tired.
The reason for riding every second day is to give those muscles time to recover and respond to the demands you're putting on them. They will grow stronger, if you don't over-stress them and give them time.
As they get stronger, you can increase the distance, by say, 10% until you are comfortable with the new distance. Don't over-do it. If you injure yourself then your whole cycling plan will get set back by months. So do monitor how you're going, talk to the people at the shop, and definitely see a doctor or physiotherapist if you think something more serious is happening. (Of course, none of us here are doctors, so don't even imagine that this is medical advice!)
In a couple of weeks you should be back to 10 miles again, and be ready to ride further or faster or every day.