It's a great sport!
Must see races: Monaco, Spa (Belgium, my fav due to the track), Melbourne (Downunder), Monza (Italy), 1st race at Singapore at night this year was cool and well organized - look for that too, and yes, Brazil is always great.
I also like the European when it's at Nürburg and this year's in Valencia was interesting as a change. I have a fondness for the Hungaroring because I was there 2005 and happened to be staying at the same hotel as Alonso - they know how to party!
Things to understand & learn:
- strategy, when to pit, tire selection etc
- handling / balance of the cars and set-up for a particular track
- aerodynamics - as above
- tires, huge and underappreciated factor, as above track contingent
- the annual drivers' waltz: who's going where - akin to free agency in the NFL
- it's good to know some history, the great drivers like Senna, Villeneuve, Prost, and all-time most frequent World Champion, 'Schumi' (doesn't resolve the question as whether he's the greatest - so much depends on the car in the modern era)
Up and coming driver: Sebastien Vettel, youngest ever winner in F1- just this year.
The links above are good references.