Based on my own past experience with a manual transmission, it's sounding like you're not getting full engagement of your clutch. Depending on your make/model, there are a couple of reasons this could happen:
- the connection between the clutch pedal and the clutch has changed / aged / failed.
- Some cars have a cable that connects the end of your clutch pedal arm to a fixture on the clutch to push it in and disengage the gearbox from the flywheel. If the cable stretches over time, or if the fixture on either end slips or bends, then the pedal may end up feeling floppy loose when fully out, and then not be able to pull/push the clutch plate far enough to disengage the tranny from the flywheel.
- Some cars have other direct mechanical linkages from pedal to the clutch, which might have bent, slipped, etc.
- Some cars have hydraulic clutches. Same idea, but you push the pedal, it pushes a hydraulic piston on one end of a tube that pushes another piston at the other end, which pushes in the clutch. If the hydraulics are on the fritz, you don't end up pushing the clutch in enough, and the tranny doesn't disengage.
- If you have mechanical linkages, the trick then is to start at one end and work to the other to see if something has slipped, stretched, broken, worn, or the like. If you Google 'How to remove clutch in 'Make Model Year', you'll probably get plenty of Youtube clips or Instructables which will start with the pedal-to-clutch linkage.
- If you have a hydraulic clutch, then it's possible that you're low on hydraulic fluid, or have formed a bubble in the line. It's much like not having your brakes bled. You press the pedal, and a bunch of effort goes into compressing an air bubble instead of actually moving fluid. There will be a reservoir in the engine compartment for the clutch, in this case, which can be checked for proper level, topped off if necessary, and bled (procedure will depend on specifics of your car.
- If the connection is OK, then you could be looking at wear on the clutch plate or on the flywheel. Getting that much wear on a flywheel is way less likely unless you do a lot of Baja sand desert rally driving or high horsepower parking lot burnouts. Assuming for now that isn't it, the clutch plate is a wear surface, just like brake pads. If it goes through enough shift cycles, it can wear so thin that a full pedal press can't push the now-thinner plate enough to disengage the gears. That, alas, means getting the clutch replaced. Ranges from not to bad to heinous, depending on the car - older ones tend to have a lot more room around the engine to work.
- It doesn't sound to me like it's likely the gearbox, especially if it's happening regardless of speed or particular gear.
Depending on your DIY experience with car repair, some of this you could probably do. The clutch will be a bit of work even for a veteran mechanic.
If you want to at least take a stab at the cable or hydraulics, I highly recommend the shop manuals made by Haynes
Haynes is the information resource for the DIY enthusiast
They served me really well through a series of earlier cars when I had more time and less money. They are based on full tear-downs and re-assembly of the vehicles, so they tend to have a lot of practical info on not just what to do, but the tools and even body position you'll need to do it. Most of the manuals are $20, which you'll likely earn back the first time you do something yourself on the car.
Hope this helps, and happy holidays.