Hard Shifting

My 2001 5 speed Rio has been getting progressively worse for shifting into gears. When I got it last year, first gear was always hard to engage from a stop (it seemed like there was something blocking it from going into gear and I had to really press the clutch pedal down and "clunk" it hard into gear). Now second and fourth are hard to shift into while driving. When I couldn't get the car into first gear at a stop (if I had it in Neutral) I used to be able to put it in second and at least be able to get the car moving. Now I can't get it into second at a stop and have to temporarily turn off the engine and then put it in first and then start it again in order to go from a stop. Also, when I can shift it into gear the stick seems to have a lot of play in it when I try and move a little it while engaged.

The gears never pop out or even seem to slip much. So is this not my clutch, but maybe some kind of cable adjustment? What could be causing this?

Classic symptoms of a worn out clutch. As the clutch disc wears, the pressure place moves closer and closer to the flywheel which moves it away from the throwout bearing. The farther from the throwout bearing it is the less it disengages the pressure plate which keeps the clutch applied enough to not let you shift easily. You might be able to adjust the cable up and buy some time but you are probably going to need a new clutch.