Proper Coin Handling (Need Advice)

Wear gloves when handling the coins. There are dedicated cotton ones for this purpose. If you don't have a pair you can use powder-free latex or vinyl gloves. Note: the "powder free" is not optional - some glove powders have chemicals that will harm your coins in time, and they will be on the outside of the glove too. And as you said, handle only by the edges.
Be very careful removing the staples. I use the small blade of a Swiss army knife - it's exactly the right size. I pry up the prongs on two staples, flip the 2X2, then ease the staples out. Toss them in a trash can before removing the coin from the 2X2.

I used to do what Hondo does, but now I use an X-acto knife to cut around the edge of the circle on the Mylar, usually first pushing the coin to one side, before making the cut, I only need about a 1/4 or 1/3 around the circle before I can push the coin out from the other side - always onto a towel or one of my jewelry trays (padded surface). No chance of damage by staple that way.

I also never toss them into my own trash, after I'm sure there is no identifying marks on them that can trace the holders back to me, I put all the empty holders into a small paper bag and toss it at the gas station trash can or Starbucks, somewhere I am not known, nor a regular customer. Don't need someone to see all the coin holders in my trash can and think the house is full of rare coins. Better safe than sorry.

I always try to find the part of those cardboard flips with staples, that I can open. Then, very carefully, bend it very wide, and remove the coin straight up without touching any staples. That is my method but it does take a steady hand and practice.

@tee, thx for including that picture. While it is a little out of scope for your question, I would add to all of the excellent advice above by mentioning that you should flatten the staples on your 2x2s. Otherwise, there is a small, but non-zero chance that the backside loops of a staple could scratch another coin when you have these 2x2 stacked.

Pleased to see I am not the only person who does this.

For my empty 2x2's I have a nice heavy duty cross cut shredder I put any old no longer needed 2x2 into and all data is shredded It even eats up staples like nobody's business

Jbuck, I don't shred flips, but I definitely shred all my incoming ebay and USPS envelopes for the same reason. I don't need any troublemakers deciding that I might have expensive coins in the house. I also used to send mail but putting the flag up on the mailbox, but after our neighborhood had the first incidents of porch pirates, I started taking mail to the post office indoor drop slot.

I am sometimes forced to cut flips to free purchased coins—I learned how to do that here—but once a kind CCF denizen enlightened me about piercing the Mylar, that became my preferred method.

Don't even get me started on the sellers who staple plastic flips! =P

Forgive my lack of knowledge, I'm still fairly new to this. When you say "Flips", what exactly are you referring to?