As the 360's my core collection focus, I think I can give a few recommendations.
First things first, you need a hard drive. Besides being required for many major late-era games, installing any game improves performance, loading speeds, and strain on the system. Skyrim, for example, will make your system sound like it's trying to kill itself if you don't install. You'll also need it to be able to download any major piece of DLC or Arcade title.
Another thing to note is Microsoft's Games for Gold program. If you have Xbox Live Gold, you get 4 free games a month, 2 for the One, and 2 for the 360/Original Xbox. While you can only access the One titles while you have a Gold membership (but can lapse and resubscribe), the others are yours to keep. The current free 360 game is Far Cry 2.
Anyhoo, on to the games. From my collection, I'd recommend the following:
Adventure Time: The Secret of the Nameless Kingdom
Alpha Protocol
Assassin's Creed Series
Batman Arkham Series
Bayonetta
Beautiful Katamari
Blue Dragon
Borderlands Series
Deus Ex: Human Revolution
Dishonored
Dragon Age Series (Inquisition is best on One/PS4)
Dynasty Warriors 7
Dynasty Warriors 8
Dynasty Warriors Gundam 2
Earth Defense Force Series
Eat Lead: The Return of Matt Hazard
Fable Series
Fallout 3
Fallout New Vegas
The Godfather
Halo: Reach
Kingdoms of Amalur
Lego Batman
Lego Star Wars: The Complete Saga
Lego The Lord of the Rings
Lost Odyssey
Marvel Ultimate Alliance series
Mass Effect series (Games must be played in order and with the same character for best effect)
Mercenaries 2
Minecraft
Mini Ninjas
Mirror's Edge
Naughty Bear
Overlord series
Portal 2
Red Faction: Guerilla
The Saboteur
Sid Meier's Civilization Revolution
The Simpsons Game
SoulCalibur series
South Park: The Stick of Truth
Star Wars: The Force Unleashed series
Tales from the Borderlands
Tales of Vesperia
Viva Pinata series
The Walking Dead series
The Witcher 2
The Wolf Among Us
X-Men Origins: Wolverine
Arcade Titles:
Brothers - A Tale of Two Sons
Castle Crashers
Dust: An Elysian Tale
Geometry Wars series
N+ series
Peggle
Penny Arcade Adventures Series
Portal
Puzzle Quest series
Splosion Man series
Super Meat Boy
The Maw
The Secret of Monkey Island Special Edition
Torchlight