We are First Impressions
There's a range of tools and techniques required to successfully excavate fossils without damaging them, including an X-ray device (to detect what lies underground), a hammer (to break down rocks and soil), and a drill (to chisel away smaller pieces of earth). Blowing dust away (using the DS's microphone, a la Spectrobes) is also an important part of recovering pieces for your dinosaur army.
Beware, however, for if greenish cracks appear on the fossil, that means that it's getting damaged, and could break. We'd recommend reading Siliconera's write-up, as it actually sounds like quite a tense (and delicate) process!