The Internet is a constantly evolving environment, and keeping up with the best tools to use is a constant challenge. At this point in time, I follow Twitter on three different kinds of hardware: my desktop computers, my tablet computers, and my smartphone. Different software will work better on some hardware than others, but these are my current favorites. What works best for you will most certainly be different, depending on your hardware, Operating Systems, and your personal preferences.
1.) Of the six desktop computers I use regularly, the Meerkat from System 76 running the Ubuntu 12.04 Operating is the one most often used to track my twitter. And on that computer I most often use the Hotot client, shown running in the screengrab above.
2.) When out and about with my smartphone, I find the Twitter for Android app to be the most effective.
3.) And on both my tablets, Kindle Fire and Nexus 7, TweetCaster serves my needs best.