Looking at the feed you should see if it's possible to exclude any tweets that begin with "RT" (retweets) or the "@" (replies). This way only posts you write should show in the feed, would make it much cleaner as without the full conversation contexts those other messages make no sense.
The GitHub activity feed is pulling in every comment you make, see if you can filter that to only show pull requests you've created to showcase your work. Maybe also include commits to your own projects if the commit length is more than a certain number of characters to filter out the small updates.
Heh, I should be clear: I consider the aforementioned Tweet/RT issue a UI/UX thing, it's otherwise working as designed.
I'm perfectly happy showing my @replies, rando GitHub activity, etc. Nobody is actually reading that sidebar, short of a select few - I like having that junk indexed in search engines attributed to my own domain.
The GitHub activity feed is pulling in every comment you make, see if you can filter that to only show pull requests you've created to showcase your work. Maybe also include commits to your own projects if the commit length is more than a certain number of characters to filter out the small updates.