It's tough to answer what an astronomer does every day since different days we do different things. We typically divide our time between 5 or 6 different things: Working in our offices, traveling, teaching, reading, writing, or observing at telescopes. Astronomers who focus on theoretical astronomy, though, aren't typically involved in observing.
Working in our offices: This usually involves doing administrative things for our institutions (though we generally try to limit the time spent doing this), or working on our research. As far as research, this could mean a wide variety of things, from calibrating and analyzing data, to running numerical models, or testing theories, or any number of different things. However, we do wind up spending a large portion of our time in front of our computers. Familiarity with computers and programming is a necessity in an astronomy career.
Traveling: If you like traveling, astronomy is the career for you. We go to a lot of conferences which are held all across the world. For example, every year there are the AAS (American Astronomical Society) and IAU (International Astronomical Union) meetings which are held at different places each year. Typically an astronomer will go to several conferences a year. The amount of trips is usually dependent on the amount of money that the astronomers have available for travel from their institutions or their grants. Most of the time we will give presentations of our research at these conferences. These conferences provide an excellent opportunity to meet other astronomers and see what kind of research other people are working on. And since there are fewer than 20,000 astronomers in the world, you do sort of get to know just about everyone in your field through these conferences. Often times astronomers are invited to give individual talks at other institutions. Most universities (with Astronomy departments) hold weekly meetings where they bring in scientists from other universities to talk about their research.
Teaching: Most astronomers teach astronomy and physics courses at colleges or universities. Just how much time we spend teaching varies from person to person, but usually it's a considerable commitment in our time (especially for younger astronomers who are teaching courses for the first time). Even as PhD students most of us at one time work as TAs (Teaching Assistants) as part of our stipend.
Reading: We spend a lot of our time reading, either learning new subjects, or just keeping up with major developments in the field of astronomy. There are many different publications like the Astrophysical Journal for example, and we need to keep up to date with current research and developments. Many of us spend the first chunk of our day checking out the new astronomy papers that have been uploaded on the arXiv, an open-access archive of new journal articles.
Writing: We also spend a lot of time writing papers and proposals. We need to write and publish papers in order to display, and share, our research and results. Good writing skills are necessary to be able to put forward one's thoughts clearly and succinctly. We also need to write grant proposals to get funding from various sources for our research, and we also need to write observing proposals to get permission to use different telescopes and facilities to further our research.
Observing: Observational Astronomers often have to go to various observatories to carry out their research. These observatories are located all across the world, from Puerto Rico, to Hawaii, Europe, Australia, Chile, or even the South Pole. Most observational astronomers are well-traveled. However, many astronomers work on purely theoretical projects and do little, if any, observing.
How our time is divided between these various activities varies from person to person and what kind of research we are doing at the time. It is safe to say that Astronomers spend an awful lot of time away from home, but we also spend a lot of time in front of our computers (though I think most of us try to minimize that).
P.S. Oh and of course, we also spend some time answering questions from curious people :)