1) Follow only people you have met in person, or otherwise directly corresponded with. I make very few exceptions to this.
2) Disable all retweets. Twitter doesn't make this easy, but there are tools that do it in bulk.
3) Make sure you're looking at 'latest tweets' mode, so that you only see the content you chose with (1) and (2). Twitter very frustratingly will put you back in 'top tweets' mode periodically, so you'll have to keep an eye on this. It feels like this happens about every 2 weeks but I haven't measured.
With this setup, nearly all of the viral BS drops away. I only see tweets in which a person I have some relationship with has taken the time to actually type something. 'quote tweets' are still problematic, but it's not nearly as bad.