First of all congratulations for wanting to grow in your Jewish learning. Learning Talmud means different things. One approach is to learn the structure of the Talmud, its way of approaching issues, the various elements of logic it uses, key elements of language (the Talmud is written in Aramaic). This is best accomplished by picking a few pages and learning them in depth, with the goal of progressing towards one chapter then a full tractate. Beginners typically start with topics such as the time to recite prayers or lost and found objects, i.e., topics of practical interest. The rabbi who you learn with will likely offer his own suggestion or some options to choose from.
If you don't own a Talmud, it is likely the rabbi you will learn with will provide you with a copy of the tractate you will learn from. Artscroll has a travel edition of their English translation of the Talmud, with all tractates broken down over a number of small and light brochures, which are relatively cheap and provide much depth for further study. They also exist as an iPad or Android application and you can purchase part of a tractate only.
The other approach is to learn the content of the entire Talmud, and indeed someone who learns all of it will have incredible breadth across many Jewish topics. However this can take very long (daf yomi takes seven years) and, unless you are familiar in advance with the topics, can be a frustrating experience. Not to discourage you but learning daf yomi requires much persistence and time commitment (30-40 minutes a day minimum if you want to follow along with a shiur, closer to 60 minutes in many cases). I wouldn't recommend this to a beginner. Before learning daf yomi you might want to start by learning all the mishna (which is the foundation of the Talmud) to get a grounding in the content. See here for resources to do so.
Congratulations again and feel free to ask more questions as you progress in your learning.