First of all, check your transmission fluid level to make sure it is at the proper level. A transmission operates off of "fluid pressure" and if the fluid level is low, the fluid pressure would also be low(er) than it should be. Hence, possibly causing your problem.
If that is not the problem, it could also be a bad shift solenoid (located on the valve body - accessible by removing the transmission pan). However, there are 3 to 4 of them, depending on the transmission you have. They all have "different" purposes. It would be necessary to know which one controls your 1st to 2nd gear shift, so the right one can be replaced. A full-featured scanning tool is able to check for DTC's (Diagnostic Trouble Codes) for the transmission too and will detect if you have a bad or malfunctioning shift solenoid. If the fluid level is not the problem, it would be best to have the vehicle scanned before going any further.
The whining could be an indication of an internal problem, however, it could also be whining because of the higher RPM.
Lastly, if this is not it either, the problem would be internal (inside the transmission). Could be a bad clutch pack or something else. I ran into the same situation in a 1996 Ford Explorer once. It wouldn't shift from 1st to 2nd sometimes (or at all) - or it would shift from 1st to 3rd (skipping 2nd gear entirely). Sometimes in order to shift, the accelerator would have to be released and re-engaged a couple seconds later, after the RPM's decreased some. The problem wound up being a broken tension ear on one of the bands. If this is the case with yours and this winds up being the problem, the transmission needs to be rebuilt or replaced. Since in order to make this repair, the transmission would have to be removed from the vehicle and completely disassembled.