Here are all the things that need to be checked!
It sounds like you could have a problem with the front band as I mentioned in #5.
The best way to check this is to see if it shifts from first to third when it does shift. After it shifts, try and pull it into manual second and see if the engine just starts to rev and you don't go anywhere! Let me know the results of this test!
This could be caused by a few different things. Your upshifts are controlled by governor pressure and your downshifts are controlled by throttle pressure and here are the things you should check!
1. Check your throttle valve cable on the driver front side of the transmission. It connects to a throttle valve lever. When the gas pedal is up the lever should be all the way forward and when you start pressing the pedal it should start moving rearward, The lever has a return spring on it to return the lever to the front. Check this for binding/sticking and make sure it works freely!
2. Your governor pressure has to be monitored with a scan tool. In drive stopped the governor pressure should be 0 psi and as you start accelerating the pressure should rise. I use a scan tool along with a pressure gauge hooked to the governor pressure port on the right side of the transmission then I compare to make sure the sensor is working properly. Governor solenoids are the most common part that fails with these 42/44/46/47/48re transmissions. I always replace the solenoid along with the sensor if found faulty. They are easy to replace you just have to pull the pan off to replace.
3. Another thing it could be is a sticking valve in the valve body and the only way to repair it is to replace the valve body or have it rebuilt by a professional.
4. Another thing you could check is the output speed sensor. This sensor tells the PCM the output shaft speed of the transmission which is then used for governor pressure calculations for the electronic governor. When the output speed sensor signal is missing there isn't any way to control the governor and it will normally stay in first gear without attempting to up shift. The output speed sensor can be checked with a scan tool or an oscilloscope. You can also check the resistance with an ohm meter to see if it is open or shorted. The resistance should be around 580-700 ohms.
5. Sometimes if your transmission won't up shift out of low until you get your revs and speed way up and then have to back off to get it to up shift you could have a front band that is way to loose, broke or fallen apart. This will not allow a shift into second gear and it will go right from first to third. The front bands are the weakest part on these transmissions!
I hope this helps, let me know and we can go from there!