How to replace gear shift cable on 2003 Chevy Trailblazer?

First dis-connect the battery. Next you will need a small metric socket, Phillips screwdriver and a couple slot blade (regular) screwdrivers.
Take out the inside compartment in the console (6 Philips screws) the screws are bit hidden (all of them) you will not see screws until you take the little rubber inserts out. Lift out the compartment. Now, under that compartment there are two small screws with metric heads, you just used 1/4" socket set with two long extensions, now, you have to remove the two cup holders in front and you will find two more metric headed screws (something the Chilton manual left out).
Next, using the small star end on a nut driver handle, remove the set screw in the base of the shifter knob (driver’s side) and remove the knob and shift boot. Next, remove the console; you might have to UN-plug some of the wires. Now the end of the shift cable will be visible in front of the shifter.
Make sure the shifter in the Park position.
Jack up the throttle body and make sure it isn't going anywhere by placing blocks in back of the rear wheels. Next crawl under and locate the cable end at the transmission, and using a flat blade screwdriver, separate (pry) the end of the cable off the end of the trans lever (which has a locking ball on it) then next the cable assembly will be held in a bracket, you have to locate a clip near the end and pry it down to free the cable from the bracket. Now follow the cable to where it enters the floor, there is a rubber boot (part of the cable) that pushes out under the carpet.
You might have to pry up the pieces on the driver’s door entrance to free up the carpet. Next, you use a screwdriver (Slot head) to pry the cable off of the shifter end (should have a retaining ball like the transmission did). Pull the cable out and get ready to put the new one in.
Installation should go reverse of the removal with the exception of making sure the shifter inside is in park as well as the transmission lever.
You will notice a small clip (transmission end of the cable) with a sliding locking clip. This should be down until you get the cable hooked up then press it up and slide the lock over it. DONE!

-----------------

Gear shift cable came off where it attaches to the tang on the shift rod on transmission; at the cable end, there's a clip holding on a 3 inch black plastic piece with hole in the end that attaches to a pin on the metal tang; on the pin there is a white nylon bushing that appears broken and that let the black plastic piece to "jump off" the tang.

The cable and connections had to be purchased as a unit.

That is an assembly and I recommend you change it. The way you describe it does sound like the cable end is damaged, and it is very likely that it will come apart again. Changing the cable will not be too bad once you get the cover of the center console off.

You are best to replace the whole assembly. Unfortunately, this will be a pain because the center console needs to be removed. Here is goes.

Block the wheels to prevent the vehicle from moving.

Remove the rubber ash tray inserts and rubber mat from the console.

The floor shift control knob and boot are serviced as an assembly.

Loosen the set screw (3) or (1) for Buick models.

Push in at the base of the boot to release the locking tabs (2) from the console assembly.

Pull up to remove the knob and boot assembly from the transmission control (4).

Open the storage bin cover.

Remove the rubber mats covering the retaining screws.

Remove the 6 screws that retain the storage bin to the console.

Remove the storage bin from the console.

.

Remove the center console retaining screws.

Adjust the parking brake to the full-up, or fully engaged position.

Slide the console rearward and raise the rear of the console in order to expose the electrical connectors.

Disconnect the electrical connectors, as necessary.

Adjust the parking brake to approximately the halfway position.

Remove the center console from the vehicle.

If replacing the floor console base, remove the following and retain for transfer:

11.1.

Remove the HVAC control module (if equipped).

11.2.

Remove the armrest.

11.3.

Remove the HVAC module and duct assembly.

11.4.

Remove the cupholder.

11.5.

Remove the HVAC upper duct.

Ensure that the vehicle is in the park position.

Raise the vehicle. .

Remove the end of the range selector cable from the transmission range selector lever ball stud (6).

Remove the retainer (5) from the range selector cable.

Remove the range selector cable (4) from the bracket (1).

Remove the cable from the retainer (3) on the floor panel. If the vehicle is equipped with 4WD, remove the cable from the retainers located on the transfer case.

Push the cable grommet (2) up through the floor panel.

Lower the vehicle.

Remove the drivers side front mat and carpet.

Remove the end of the range selector cable (1) from the floor control ball stud.

Remove the range selector cable (2) from the floor shift control assembly.

Remove the tape and grommet on the shift cable from the floor panel.

Remove the range selector cable (3) from the vehicle.

Adjsutment

Ensure that the range selector cable is not restricted.

Ensure that the floor shift control is in the PARK position.

Raise the vehicle. .

Disconnect the range selector cable (4) from the range selector lever ball stud (6).

Ensure that the range selector lever is in the mechanical PARK position. (Rotate the range selector lever fully clockwise.)

Release the locking tab (1).

Slide the secondary lock cover (1) to the side.

Carefully squeeze the locking tabs (2) together to disengage the primary lock.

Pull the primary lock (1) up. Spring tension will push the end of the cable past the ball stud.

Important: If the cable end is pushed rearward past the ball stud during the adjustment procedure, it must be released and allowed to come forward of the ball stud. The cable end must then be pushed back just enough to be installed to the ball stud.

Push the end of the cable until it is aligned with the ball stud.

Install the cable (4) to the ball stud (6).

Seat the primary lock (1) by pressing into the locked position.

Slide the secondary lock (1) over the primary lock.

Be sure that the secondary lock tab (1) is securely in place.

Lower the vehicle.

Check the vehicle for proper operation.