While these techniques won't work on the iOS and Android mobile apps, they will clean up your Twitter experience on the web.
They work by adding filters to an Ad-blocker extension.
I recommend uBlock Origin or Adblock Plus.
Then you can add the following lines of "code" to hide certain types of content from Twitter.com.
For more information on how to add filters, view this mini-tutorial.
While you can hide Retweets on a per-account basis from the main timeline, Retweets will still appear on each users' profile and on Twitter Lists.
This filter will remove Retweets from these places as well:
twitter.com##svg>g>path[d="M23.615 15.477c-.47-.47-1.23-.47-1.697 0l-1.326 1.326V7.4c0-2.178-1.772-3.95-3.95-3.95h-5.2c-.663 0-1.2.538-1.2 1.2s.537 1.2 1.2 1.2h5.2c.854 0 1.55.695 1.55 1.55v9.403l-1.326-1.326c-.47-.47-1.23-.47-1.697 0s-.47 1.23 0 1.697l3.374 3.375c.234.233.5126.96.36.199s.613-.116.848-.35l3.375-3.376c.467-.47.467-1.23-.002-1.697zM12.562 18.5h-5.2c-.854 0-1.55-.695-1.55-1.55V7.547l1.326 1.326c.234.235.542.352.848.352s.614-.117.85-.352c.468-.47.468-1.23 0-1.697L5.46 3.8c-.47-.468-1.23-.468-1.697 0L.388 7.177c-.47.47-.47 1.23 0 1.697s1.23.47 1.697 0L3.41 7.547v9.403c0 2.178 1.773 3.95 3.95 3.95h5.2c.664 0 1.2-.538 1.2-1.2s-.535-1.2-1.198-1.2z"]:upward(article)
Remove Other Sections
You might want to clean up your Twitter web experience even more.
A set of filters hosted on GitHub called CleanYourTwitter hides the following:
Just copy and paste the code from here into your Adblocker filter list.