Why is my vehicle not shifting pass third gear

The first thing you'll want to do is get a scanner on the car and look for transmission codes - all systems are a must here. Either buy something like this scanner, or pay a shop to scan for codes: https://www.amazon.com/dp/B0DF55W14J

If you do not find any concerning hard stop codes, confirm the symptom and see what happens when you drive it under a load of 60-75mph, give or take. Find a road were the limit allows for it, or an empty highway that isn't on a ticket-friendly street or empty back road. As soon as you're comfortable, put your foot down with some caution (especially given the trans is already a potential issue) and drive it as hard as you safely can, while watching for issues like clunking. If it seems like it is clunking, STOP - THE TRANS IS POTENTIALLY AT FAULT AND YOU WILL MAKE IT WORSE! If you are in Georgia or NC, do this on a track or backroad unless you can afford a ticket.

The problem is if the trans is bad, you'll probably need to pay a shop to rebuild it unless you can find and install a good reman yourself. These are FF cars, so the engine has to come up, unlike the G8 and SS (Holden Commodore, which is an FR layout). The shop costs will total out this car being a typical FF GM of the early 2000s.