2002 Toyota Celica GTS, 6-speed manual: Clutch operation/condition normal, shift linkage shifting into all gears, engine starts/revs as does speedo but car doesn't move. No visual signs off case damage or leaking fluids. No strange noises present. Suggestions?

manual or auto would be a big help
If manual check clutch operation
cable clutch --replace cable
hydraulic clutch ---replace clutch master cylinder
Auto check for torque converter clutch servo/solenoid problem or TCM or selector linkage

They probely need adjusted but if you bought it new and it started that go back to the dealer have them fix it .

If your car is manual, clutch not releasing completely, loose or damaged shift linkage, insufficient transmission oil or oil is in poor condition, sticking or jamming gears, worn or damaged striking rod. Autozone, Advanced Auto, or O'reilly can scan/read your check engine code. Speedo faulty or vss speed sensor no longer works.

Have you had problems with clutch slipping up to this point? Like engine revving up and no change in speed while driving. Like going into 4th or 5th gear, letting clutch out, giving gas, and feels like no response? Engine just revs up? That's a slipping clutch.
If that's your problem, you need a new clutch.

The reason is that the gears are turn with the engine running, like the clutch is not disengaged. So that is what you need to look at. the clutch or clutch pedal linkage/ hydraulic system. One of these is not letting the transmission slow down enough to engage a gear.

There may be several reasons for this. 1st, low transmission fluid levels can cause the gears to grind while shifting due to lack of lubrication. I would suggest checking the fluid level, there is a plug located on the side, in about the middle of the transmission which when removed should slowly seep oil from it. If it doesn't then you will need to add the recommended transmission fluid.
2nd, a maladjusted clutch can cause the same problem. Even though you have the clutch pedal to the floor, the clutch pressure plate may not be completely disengaging from the flywheel. This will cause pressure on the gears and synchronizers while trying to shift. I suggest having the clutch adjusted by a mechanic if you have never performed this function yourself. An inproper adjustment can severely shorten the life of the clutch.
The 3rd option may be a warn clutch component like the slave cylinder if equipped, the throwout bearing or even the pressure plate itself.
A visual inspection can be made of the clutch by removing the inspection plate from the bottom of the bell housing to view the flywheel and clutch. With someone depressing the clutch you should be able to see the clutch moving off the flywheel.

possible selector linkage adjustment , loose or shifted

Automatic transmission are complex systems. The shifting of gears is based on a number of things like bands, speed, acceleration, deceleration, manifold back pressure, RPMs, temperature, etc.
Sometimes the electrical ground connection for the transmission can become corroded which causes poor shifting.
Since you are getting an error code, you should get a readout for it. Many auto supply store like Advance Auto will loan a meter for you to get a readout. Call up some stores to find one.

just ram the shifter in your booty,.....duh

Grinding going into third gear only may indicate a worn syncronizer ring. There are no adjustments on the shift cables, so I doubt highly that changing the cables will help you, since you are talking about 3rd gear only. Please note: This issue may be still under warranty, if your vehicle has less than 60,000 miles and is not 5 years old. Replacement of 6-speed internal parts is best left to a pro.