In no particular order:

• Basic Fantasy RPG
  
• Savage Worlds
  
• Dungeon World
  
• Mouse Guard Check under other purchase options. You can get it for about $19.40.
  
  You can't go wrong with any of them, but gun to my head, I'd say look into Mouse Guard or Dungeon World.
  
  

  Mouse Guard
  

  Mouse Guard...is just down right fun. While combat is certainly part of the game, it's got a heavy emphasis on problem solving in encounters over straight up murderhobo-ing your way through the game. It's my go to game for one shots. Perfect for when you need a break in the middle of a long campaign in another system, or if just not enough people show up for a session in your primary campaign. Not that you couldn't run a long campaign in Mouse Guard, that'd be rad.
  
  

  Dungeon World
  

  Dungeon World is also another great game for one shots, IMHO. Need to work on your improv? Play Dungeon World. It's a very narrative driven game that heavily encourages collaborative world building with your players. Additionally, if you discard all of the rules on classes and combat, the rest of the system is an amazing compliment to any other game system you want to run. It's GMing section is honestly the missing manual for D&D or any other system you want to run. And hell, Fronts are a great way to organize an adventure or long campaign.
  
  

  Edit: A few other options that came to mind
  

• Fate: Core System, or Fate Accelerated Everyone seems to either love or hate the system with no in-between. If you've played the Dresden Files RPG, you're familiar with the system. Fate Core was derived from the Dresden Files RPG.
  
• Bubble Gum Shoe This one is a lot of fun. Runs on the Gumeshoe system. Kind of an innocent system. Think Scooby-Doo mysteries, without the monsters. If you want something grittier, take a look at Trail of Cthulhu or Mutant City Blues.
  
• Monster of the Week This one is a guilty pleasure of mine. Game is exactly what you'd expect from the title. If you like episodic shows like Supernatural or Buffy, this is for you.
  
• Fiasco Haven't played this one myself yet, but it looks interesting. This is a game that doesn't require a GM and is entirely improv. Looks great. Requires six sided dice.

It can be done, but it'd be tough. I once played in a treasure hunting game where there was a competing group going against us. Sometimes we got the treasure, other times they did.

I'd recommend writing a really episodic campaign. Make sure each session is a self-contained story. It will need plenty of paths to the same session climax. Maybe treat the campaign like a string of one-shots with all the same reoccurring PCs.

A good game for two groups, same campaign would be Monster of the Week. It's intended to emulate a Supernatural, Buffy the Vampire Killer style of play. You can have themes that carry over week to week, but each session is it's own story. The chapter about writing your adventure notes is top-notch. I've heard of GMs running the same MotW session for several different groups before.

Man softcover from approved retailers is really hard (no sarcasm), so many good games only come in hardcover. If you give me more information about his preferences I could help you narrow the list down.

https://www.amazon.com/Esoterrorists-RPG-2nd-Robin-Laws/dp/1908983523/ref=sr_1_1?s=books&ie=UTF8&qid=1494289945&sr=8-1&keywords=Esoterrorists

https://www.amazon.com/Blue-Rose-AGE-Romantic-Fantasy/dp/1934547743/ref=sr_1_2?ie=UTF8&qid=1494289980&sr=8-2&keywords=Blue+rose+rpg

https://www.amazon.com/Microscope-Ben-Robbins/dp/0983277907/ref=pd_rhf_dp_s_cp_9?_encoding=UTF8&pd_rd_i=0983277907&pd_rd_r=Z3RJ2JPXJQYDRYA06S2Y&pd_rd_w=hHGH4&pd_rd_wg=W5G5p&psc=1&refRID=Z3RJ2JPXJQYDRYA06S2Y

https://www.amazon.com/Bully-Pulpit-Games-BPG-005/dp/1934859397/ref=pd_sim_14_3?_encoding=UTF8&pd_rd_i=1934859397&pd_rd_r=JBAWXXB309ERX5KH0PXT&pd_rd_w=Eunmp&pd_rd_wg=OTd2y&psc=1&refRID=JBAWXXB309ERX5KH0PXT

https://www.amazon.com/Monster-of-the-Week-EHP0009/dp/1613170920/ref=pd_rhf_dp_s_cp_2?_encoding=UTF8&pd_rd_i=1613170920&pd_rd_r=WFVTZB7V8R3S7S8F462W&pd_rd_w=HYjgo&pd_rd_wg=K0zZT&psc=1&refRID=WFVTZB7V8R3S7S8F462W

https://www.amazon.com/Evil-Hat-Productions-EHP0002-Accelerated/dp/1613170475/ref=pd_sim_14_8?_encoding=UTF8&pd_rd_i=1613170475&pd_rd_r=JBAWXXB309ERX5KH0PXT&pd_rd_w=Eunmp&pd_rd_wg=OTd2y&psc=1&refRID=JBAWXXB309ERX5KH0PXT

https://www.amazon.com/Urban-Shadows-Softcover-MPG007-Truman/dp/1987916166/ref=pd_sim_21_2?_encoding=UTF8&pd_rd_i=1987916166&pd_rd_r=DKP8DTTM0N8MT2HDRFPZ&pd_rd_w=i0nmQ&pd_rd_wg=KPg28&psc=1&refRID=DKP8DTTM0N8MT2HDRFPZ

https://www.amazon.com/Kingdom-Ben-Robbins/dp/0983277915/ref=pd_rhf_dp_s_cp_13?_encoding=UTF8&pd_rd_i=0983277915&pd_rd_r=A9ZZTCT4BD1KY3DA9ZYW&pd_rd_w=jRFtT&pd_rd_wg=biDau&psc=1&refRID=A9ZZTCT4BD1KY3DA9ZYW

https://www.amazon.com/Bully-Pulpit-Games-Durance-Playing/dp/0988390906/ref=pd_sim_14_26?_encoding=UTF8&pd_rd_i=0988390906&pd_rd_r=T0HFDYG416QT8CZBM6C7&pd_rd_w=ljUZO&pd_rd_wg=hK410&psc=1&refRID=T0HFDYG416QT8CZBM6C7 (I think)

Character playbooks for pbta might be hard to print off.

https://www.amazon.com/Havenshield-Complete-RPG-Rulebook-Myers/dp/154258356X/ref=sr_1_7?s=books&ie=UTF8&qid=1494291018&sr=1-7&keywords=Rpg (this seems like its worth a look)