Try it again on an instant action mission with your jet either warmed up and lined up on the runway or even better, already in the air. All relevant switches and stuff should be turned on and working.
Then I want you to do the following steps once you are up in the air and fairly straight-and-level:
1. Check what AP mode you are on by VISUALLY checking the toggle switch in the in-game cockpit. Verify if it is on PATH, ALT/HDG, or ALT. I think it defaults to ALT/HDG, but regardless, verify which mode it is on.
2. Change your AP mode to whatever mode you want it to be. Visually check again in-cockpit that the toggle switch is in the appropriate position.
3. Configure your aircraft so that the AP **should** come on. For example, if you chose ALT, be nose-level and bank about 45 degrees. If you chose ALT/HDG, be nose-level and wings-level.
4. Engage the AP. You will NOT hear a confirmation that the AP has come on, but should be displayed on the left lower side of the HUD. If you move your aircraft to override the AP, you will hear a warning.
I hope that helps you out.
BTW, not seeing the FPM/TVV on the HUD may mean you are going too slow and are about to stall? If you are going full-throttle, but still cannot engage AP and still cannot see the FPM, then something is wrong with your aircraft.