Hopefully I'm not too late for the "First Game" bandwagon.
If I'm remembering correctly, I received my NES as a joint gift from my godparents (an aunt and an uncle, both from my father's family) in 1989. I would have been four years old at the time.
It was either in November or December of that year, as my birthday is in November, and Christmas is in December. I'm leaning towards it being a birthday present as I don't recall it being given to me with a whole bunch of other family members being around.
It was the NES Action Set:
So
technically my first game was the Super Mario Bros./Duck Hunt combo cart.
However, the first game I received as a gift was Chip & Dale: Rescue Rangers, which I distinctly remember my father giving me when he came home from work one day. I don't think this was for a specific reason or celebration.
As far as the second game I received goes, I honestly couldn't tell you. I still have all my carts from when I was a kid with the exception of a few that I sold over the years that I didn't have any personal attachment to. (Jaws, for instance, which I bought from a neighbor and didn't care for for obvious reasons.)
I do distinctly remember picking out Kirby's Adventure though. I have vivid photographic memories of all the NES boxes lined up on a pegboard behind glass at our local KMart. I also have fond distinct memories of the first time I finished the game.
The last NES game I remember picking out when they were still sold new in stores was Mega Man 5, and it shows because it's the one I have in the best condition. (box and all)
I later received The Jungle Book and Space Shuttle Project as Christmas gifts, but these were both games I requested for Christmas and didn't personally pick out behind from a case.
I'd love to track down the exact model of TV I used to play my NES on when I was a kid someday. It was a 13" Samsung with big ol' VHF and UHF knobs on the front. I found it last year on eBay but after it sold I haven't seen a trace of it since, and unfortunately I can't think of the model name to save my life. Bah.
Edit: After wracking my brain for the last hour I feel like the model name as "Symphony", though I could be mistaken.
It looked damn close to this one here, though I remember the volume knob being on the bottom right...you would pull it out to turn it on and then twist it to change the volume.