Easy. Borrow a SIM from a friend who has an account with a different wireless service provider and see if it works in your phone or not. If it works, and your phone thinks it is his (or her) phone with that phone's number, then your phone is already unlocked.
But if it creates some sort of error message and doesn't work, then your phone is locked.
There are two main companies that offer GSM service in the USA. AT&T and Cingular are now merged into one company, and the second company is T-Mobile.
If your service is with, eg, Sprint, Nextel, Verizon, or just about any other US wireless company, then you do not have GSM service and your phone can't be unlocked.
I've unlocked my phone, but when I turn it on, it still shows the name of the previous wireless service
This logo (called a 'splash screen') has been programmed into your phone by the company that sold it to you. It doesn't mean anything about which company now provides you phone service.
Unlocking your phone removes the electronic restriction that prevents your phone from working with other SIMs from other wireless services. It doesn't change your splash screen.
The good news is you can get rid of the splash screen without needing to unlock your phone. See if you can find in your phone's manual how to change the splash screen, or consider calling the phone manufacturer and have them walk you through it.