Okay, that's a start.

Gently flip your machine onto its back so you can see its underside. Then slowly press on one of the keys - you're not trying to get it to hit the platen, you just want to move it through its entire stroke slowly. As you're doing this, observe what's going on underneath the machine. The following should be happening:
1. This linkage should move upward.
2. The shaft should then twist with the U-shaped portion moving toward you.
3. The top of this silver assembly should pivot toward you.
4. The escapement wheel should turn clockwise.

Let me know which, if any, of these steps is not happening. Oh, and if you post photos it would help a lot as we might be able to spot something obvious just from looking at your machine.

Well, that narrows it down quite a bit. Regarding #1, I meant upward relative to the photo (I didn't want to confuse things); regardless, it's moving in the right direction. 

We now know the problem is in the area of the escapement. I'll have to take more photos (I'll try to get to it tomorrow) to troubleshoot the problem even further. On the other hand, now that you know which part is supposed to move, and how it's supposed to move, you could take a very close look in that area and try and find the problem. There's a small arm that pushes down on the piece of metal that is running parallel to the arrow for #3. If that isn't moving down (relative to the photo) then the problem is in the linkage between it and #2.

I have a very similar problem with an older model (from the 1940's, I think).  All of the keys work, but the carriage does not advance unless I apply very slight pressure to the carriage at the same time I strike the key.  Any advice how to fix that would be greatly appreciated.  Thanks!