Pat Kirwan wrote a book called "Take your eye off the ball" partly about how to notice what's going on during the game.
Offense
Pay attention to personnel packages. Watch the subs as they come on the field and if you recognize the positions based on the jersey numbers then it gets easy to track what personnel we're running:
- 11 (1 RB 1 TE 3 WR) base
- 12 (1 RB 2 TE 2 WR) +run -pass
- 10 (1 RB 0 TE 4 WR) +pass -run
- 02 (0 RB 2 TE 3 WR) qb mobility
Next, look at the defensive coverage vs. your personnel package to see where you have the advantage. If they are playing cover two vs. your 12 personnel then they're daring you to run on them. If they are playing cover 1 vs. 10 personnel then they're daring you to take shots.
Then it mostly comes down to the QB play - do they adjust plays when needed, make quick decisions and throws, and avoid making mistakes. Decisions are more important to me than how pretty the throw looks at this stage because the coaches have tons of times to watch throws in practice. This is the first chance to see decisions that matter vs. a real opponent. Pass protection is a QB play, OL coordination and play calling where you can see if the entire thing is working but usually not whose fault the breakdown was unless it's the RB getting pancaked or something.
For receivers my main emphasis is on whether they catch the ball. Everybody on an NFL roster can run fast. Some have the ability to make difficult catches while most don't. You can always blame ball placement or contact on the part of the defender, but the good receivers in the NFL are very highly paid because they tend to come down with the football when it's thrown in their direction. Give me a receiver who makes the contested catch when it matters over one who looks open deep but then can't adjust to an imperfectly thrown ball in the air, or catches a bunch in practice but then drops them on Sundays.
I don't think you can evaluate the running game very well live, other than just whether it working well or not as a complete package. There's a lot of interconnected parts and if it isn't working in unison then it's easy to blame the wrong thing.
Defense
Pay attention to DBs subbing in and out. More DBs = better coverage on the back end at the expense of run defense: 4 DBs base, 5 DBs nickel, 6 DBs dime, 7 DBs bandit. Once the defense lines up count the safeties deep to see if it's cover 1, 2 or 3.
In a pass rush situation look at who is getting reps on the Cheetah package (2 OLB + 2 DE + 1 DT): Harris/QJeff/Ford/Nwosu/Taylor are the starters. What kind of pre-snap movements did we make to confuse the QB, did we blitz, and did we generate pressure? If they have a receiving TE then how are we covering them - strong safety, nickel corner or outside linebacker?
In a run defense situation look at whether the DEs are playing 1 gap (lined up on the inside of their guard) or 2 gap (lined up over top of the guard), and how the ILBs are lined up behind them to support those gaps. 1 gap DEs should be penetrating upfield in those gaps to try to make contact with the runner early, 2 gap DEs should be "guarding" both of their gaps.
CB play is straightforward; were they pressing or playing off man and did they stay with their guy. Was the CB handing the receiver off to a safety as part of zone coverage, or if it was man coverage did they keep inside leverage throughout the route.