There are several third party tools out there that would enable you to enhance your tweeting experience, below is the editor's choice of two of the best web tools to use with your Twitter. As a teacher, using these tools will help you expand and enrich your professional learning experience by enabling you to follow like minded people, create lists to crowdsource information around a particular topic, read your Twitter stream in a magazine-like format, and make the best of the networking potential of hashtags.
TweetDeck is by far the best of them all. I use TweetDeck app on my Mac and it works wonders. It lets you create lists and collections around a specific topic. I have a list titled Educational Technology to which I added a number of popular EdTech Tweeters. TweetDeck allows you to track Twitter feeds of the people you follow in real time and has a notification column where you can see tweets mentioning you. I would suggest that you tinker around with the different features it has to learn more about how it works.
This is a great tool for creating magazines from your Twitter feeds. To start using it you need to sign in using your Twitter account. Once logged in, click on "create a paper", give it a title and short description then choose what you want to be featured in your magazine. You can for instance create a magazine from only your Twitter feeds, or from Tweets from your timeline or use both of them in one single page. You have a lot of options to choose from. You can also change the template and themes of your magazine and customize the fonts and text as you want. I personally use Paper.ly to read Tweets on my timeline.