If you don't mind starting small you can blink quite a few LED's with an Arduino. The simplest way to start is to connect one output pin per row of LEDs and one output pin per column of LEDs. When you set one row pin HIGH (all the rest LOW) and any combination of column pins LOW (all the rest HIGH) current will flow out through the selected row pin and in through the selected column pins. By sweeping through the rows as quickly as you can, setting the columns you want lit, you can selectively light any combination of LEDs in the array.
Give it a try with a small array like 3x3 or 4x4.