How can I discover topics in a social media data-set?
You can take a look at Latent Dirichlet Allocation. In my experience this does very well without too much effort. You need to remove words that don't help like stopwords (and in your case Twitter handles and probably URLs) before feeding it to the algorithm. The only really important parameter that you need to give it is the number of topics. This will depend on your population (are these random tweets, or only tweets from a specific subgroup/hashtag?) and you need to compare some settings. What you can do is print the most important words per topic and see if they indeed do belong together.

If there are different languages in your tweets you need to deal with that beforehand, maybe classify them on language and only keep the English ones for example.