Vans called marshutni depart Yerevan for Gyumri from Kino Rosia's parking lot, above Zoravor Andranik subway station across from the National Cathedral. These are cheap, at about $5, but often crowded or uncomfortable. They are, however a great way to meet locals. Shared taxis depart from the same location and are usually faster and more comfortable. Taxis can be taken from anywhere in Yerevan or, indeed Armenia.
Another good option for a visit to Gyumri is a day tour from Yerevan. Many companies offer these, for a reasonable price, and allow you to get in a van, with other travelers and a guide, with stops along the way in places like Talin Cathedral and Harijavank Monastery. This is probably the easiest option, and may even work out to be your cheapest compared to taxis.