There's a lot of aspects to game development, including gameplay, art, storyline, and many of those aren't technical. If you do get into a technical problem that can't be easily solved, cheat! Many games have an addictive mechanic that comes from a hard technical problem you want the player to solve.
There are some frameworks for writing games, and you can get familiar with whatever you think will solve the problem.
Also, just apply to a game company! They need web people too, and it's easier to use your skills to get your foot in the door, then you can learn from the inside.
You should be able to find a lot of tutorials on the phaser site itself as well as https://www.reddit.com/r/phaser/ or https://www.reddit.com/r/gamedev/
It's a HTML5 game engine and I've used it to create some really simple games, game development is a lot of work/time however so see if you can get into it as a hobby and you will know if the enjoyment to work ratio is worth it for you.
https://www.unrealengine.com/what-is-unreal-engine-4
It is free to download, you can play with it all you want. You must agree to pay royalties (5%) if you start making money with a game you develop.