1999 Pontiac Grand Prix Ticking

When the engine is cold remove the serpentine belt and start the engine and listen for the same ticking noise. If you don't hear it the noise is associated with the accessories pullies. Inspect each pulley for damage and wiggle and to check for play in the bearings of each pulley. Spin each pulley by hand and listen for a noises.
Let me know what you find.

Alright I did as you said and didn't hear the noise with the belt off. I inspected the pulleys and none have damage to them, I also spun them by hand and none made any noise. The only ones that did have any wiggle an play to them was the water pump pulley and the power steering pulley, now the water pump pulley has very little play to it but the power steering pulley has more play to it and moves about 1mm in each direction.

Good. So the noise is related to the pulleys. Using a stethoscope (be extremely careful around moving parts!!!), place it on or next to each pulley when the noise happens and listen for the loudest point to isolate what component is causing the ticking noise (engine running).

If you still have the belt off, then check each pulley for noise by spinning by hand (it is more difficult to hear softer noises by ear only, a stethoscope amplifies these softer noises). Engine off

Where would I even get a stethoscope. Can the noise be caused by how cold it is outside, because when I first started hearing the noise the weather was about 30-40 degrees each day and for the past two days it has been around the mid 60s in temperature. During those two days I didn't hear the noise at all when I would start the car and I only heard it again when I started the car this evening and it was around the mid 30s outside. Just wondering since I didn't hear the noise during those two days when I was warm outside.

Try Google for a "mechanics stethoscope". I got mine from Harbor Freight and it was under $10. Another test I just though of was to spray water (with a spray bottle) directly on the belt when the noise is happening. If the water makes the noise go away the noise is related to the belt. To correct this replace the belt. If the belt tensioner has not been replaced in the last 60-100k miles it wouldn't be a bad idea to replace it also.
The stethoscope will help you isolate where the noise is loudest to narrow your diagnostic.