Disclosure:
I own a board game publisher and this is my full-time job.
There are plenty of new board games that are very easy to learn. Machi Koro is one of ours that has a ~4 page rulebook with less than three rules you need to remember in order to play.
Splendor, Ticket to Ride, Catan, King of Tokyo are all very easy to play games that I think will have a better play experience than "classic" games that you are used to playing.
There is a good resource here:
https://boardgamegeek.com/wiki/page/Board_Game_Gift_Guide_2015 to find new games.
I would even posit that most "classic" games are actually relatively complex, it's just that you have the rules to those games ingrained in your head so they seem very simple to play.
That said, there are a world of increadibly complex games that you can steer clear of if that is not your thing. We certainly publish those as well.
In this thread there are at least three examples of why hobby games are not growing faster, however and is why you are probably afraid of trying more complex games.
1) Someone made a non-board game player take on a game of Die Macher which is one of the most complex games that exists. Congrats friend of poster, you just ruined hobby gaming for a group of potential new gamers forever.
2) Someone threw out Kemet as a replacement for Risk. KEMET IS VERY COMPLEX. If you want a replacement for Risk, go get Small World. It's a pretty easy to play game with lots of videos. This is an example of "Alpha gamer syndrome". Kemet is not the most complicated game, so people who like complicated games think Kemet is an easy game when it is in fact very complex and will scare you away from hobby gaming forever.
3) Someone asked for a game like Catan... and a poster told them to buy Terra Mystica. DO NOT BUY TERRA MYSTICA. It's VERY COMPLEX. There are lots of games that are "like Catan". Go get Ticket to Ride. Go buy Splendor. Machi Koro. Century Spice Road.
Hardcore Gamer's, as an interested party financially to the hobby gaming world growing, please stop making new players play complex games and let them dabble in the gateway world for 6 months before deciding it's time that they play TI:3.