Well 1-3 aren't like number 4. So I'm going with the idea that this person has children under the age of 5.
 
First of all you can throw playing competitively online out the window unless after bed time. And even then you better look at getting headphones so waking up the household isn't an issue.
 
For console I'm picking 360. 
 
First game is not a game, download Netflix and enjoy letting your kids watch Looney Toons.
 
Borderlands
Puzzle Quest
Aegis Wing (free so they can see if they like it.)
Castle Crashers
Geometry Wars 2 or Rocket Riot
Madden (or applicable game for their favorite sport)
 
That wasn't 10 but in general if a friend was just starting out I'd probably get them a points card as a gift and suggest some arcade  games to start out. Especially since they like their iphone games.

If you were tasked by a co-worker or student who likes the idea of videogames, or remembers them fondly, but simply don't have the know-how or knowledge to sift through the crap that gets released month-after-month, what kind of games would you suggest just off the bat, without consideration for personal taste?
 

First, you pick who you you are addressing:

1. Parent with one daughter and one son
2. Parent with two sons
3. Parent with two daughters
4. Single adult or Married adult w/o children, age 28-40

Second, pick 10 games you would suggest. Keep these details in mind (you can pick one or combine multiple choices):

• They may or may not have enjoyed Goldeneye on N64
• They enjoy their preloaded games on their phone/iphone app store
• They may or may not have enjoyed an obscure RPG during the PS1/Dreamcast era
• Their last point of reference may or may not be a friend coercing them to try Modern Warfare, and failing miserably
• They've played with a relative's Wii once or twice
  
• They may or may not own a PS2