Is there a specific way I need to use this display?
Yes, and it is quite specific. The information must be provided by the manufacturer or the vendor since there is no generic software that works with all serial adapters.
On the other hand, your device most likely has a PIC processor on it and there is a good chance that the software on it was written by or derived from software written by P.H. Anderson back in the microprocessor stone age.
Mr. Andersen is deceased but others are maintaining his work. If you cannot get some definitive information from Sainsmart or Amazon about your specific device then a search for "phandersen" may yield some useful information.
Somehow I think you are going to need a lot more information to really use the display. For example: How do you deal with clearing the screen? How do you position the cursor? How do you implement user defined characters?
Finally got it to work using the link given! Thanks for your help!
Before I mark it as solved, I had one more question. Is this display safe to use? I noticed that if the VCC wasn't connected, the backlight tried to turn on using the RX pin... Is this a cause for concern?