Before you leave your hangout, I recommend that you turn your interest to two optional activities. First of all, it is worthwhile to talk to Alfred, thanks to which you will receive additional experience points.
[New Character Trophy: Alfred]
Apart from that, you can go to the training area and play through the newly-unlocked Combat Training that is a part of the Dark Knight Challenges. Also in this case, you will be rewarded with experience points (more about this in one of the final chapters of this guide).
Penguin's arms dealer, who you are currently looking for, is staying in the district called The Bowery, i.e. in the North-Western part of Arkham. You can get there straight away by selecting the fast travel point from the world map (Batwing Drop Point - the screenshot), or by doing that after you interact with the protagonist's airship (Batwing). After you use the option, the game will play a cutscene and you will be transported to the Coventry GCR Tower.