My main problem is my diverse interest in different subjects and twitters current inability to let me organize and follow what I like.
I'm following approx. 200 people divided 30% technology, 30% cycling and 30% friends/locals. For me, it would be impossible to imagine following more than 250 or 300 people with todays interface - because they are all thrown into each other and reading the raw feed is a clutter and mess of subjects.
You would think, considering their main goal is to get people following their interest, that they would get this part of the interface right. But the contrary - it is what is worst with twitter.
The solution (and problem) I'm hinting to is of course lists and as Facebook, G+ and virtually every other social network already have found out: people like to organize interests, people and subjects into different "buckets". Facebook had a lame interface for this many years, but does a better job now.
My point is, as an experienced twitter user, I know where the pains are and my first day in office I would make sure that the accessibility of lists were greatly improved.
The second day I would use to fix a decent conversation view and comprehensible reply scheme.
EDIT: To point out the inaccessibility of lists today, here is the general way to read up on a subject: tool-icon > lists > choose list. That's 2 clicks too many. You could also use the shortcut "gl" and spare 2 clicks, but still, it is 1 request too much and way too complex for the regular user.