Gaming is great way to bond with your partner as it's one of the few ways where you can work together towards a goal and do things together in real-time in LDR. There are countless games these days you can play with your partner. Whether it be co-op or competitive.

PC Games - Best choice due to vast selection, lost cost to buy games. No need for monthly xbox / psn live etc.. to play together.

League Of Legends - Is an easy mmo to get into, but hard game to master. Playing a Bottom ADC/Support combo is fun way to play together in LOL. Otherwise you may be doing your own things alone on different lanes, if bottom or support gets insta-locked have someone be Jungle to get ganks/support and have semi co-op feel.

Overwatch - Great game to play together and spam quick matches or arcade mode. Just join team chat if you will play competitive together ;) Nice part about Overwatch is with Custom Games you can do aim-training custom games which can be a valuable resource to someone new to FPS.

Minecraft - Chill build something together experience, be sure to play on surival it's way more fun. You can find cheap hosting on BeastNode for just few dollars a month to play together. If you SO has a bad computer look into the Optifine mod, allows minecraft to run on anything.

Apex Legends - They launched duos so you can play together, is a fun co-op experience. Lot less mechanic heavy and more (can you aim). It's arugelyl mo

Fornite - Lot less serious than other Battle Royal games, can be more appealing aesthetic for some. Just know building mechanics make it harder to learn than other battle royal games (imo).  

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Mobile Gaming - Decent choice for those who do not have a capable computer.

PUBG Mobile

Fortnite

Apex Legends

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Console Gaming - Good for people who already have consoles.

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