Transmission slow shifting from first to second?

When first starting to drive down the street and your motor or engine should change into what I call second gear, it will not always change, and you drive at a reduced speed (10-15 miles per hour) until it does change gears. However, the rpms are running close to 3,000 which I know is bad for the motor. I took it to an Advance Auto, and they checked the sensors. They gave me a listing of the sensors that needed to be replaced. I took it to a transmission repair shop and they replaced the bad sensors. It was still doing the same thing after they replaced the sensors, so they told me that the problems were not in the transmission. They said to take it to my regular garage and gave me a list of sensors that could be bad.

Do you have any thoughts, clues or suggestions on these problems?

Thanks

Stumped in VA