Ok, I searched the Android forum and I found a lot of hits on digging deeper but the only thing I learned is that many people seem to understand how it works and I still do not!

What does digging deeper cost?

The rule book says:
"Upon digging deeper, the player advances his hero marker so it is next to the next-most valuable pile of puzzle pieces."

So, I'm thinking that's it. You collect one piece of evidence and you can pick a puzzle piece for which your hero mark is already at OR you can advance your hero marker up to the next most valuable pile. But if you advance your hero marker, you cannot take a puzzle piece. You first have to spend time moving around the board and collect another piece of evidence. Then with this 2nd piece of evidence you can uncover a part of the conspiracy by picking up a puzzle piece at the new pile your hero marker is now sitting next to.

Is that correct?

TIA!

Yeah, that's it. One other important point is that after playing a puzzle piece, your marker moves back to the lowest pile. (Which is why digging deeper is often not worth it...even though the higher piles have much more powerful effects, each piece from the highest pile would require you to follow up three pieces of evidence (unless one or more of the lower piles have already been emptied))