How Do We Get Our Swing Set To Stay Securely In The Ground. The Stacks Holding It Down Eventually Come Up And It Seems Unsafe To Me. Is There Someone Who Has A Trick To Really Secure A Well-used Swing Set?

I went to our local hardware store (a GREAT hardward store like they don't make any more) and picked up 4 small S-hooks, 4 small turnbuckles, and 4 12" auger tie-downs like a miniture version of what would hold a mobile home down in a hurricane. I also got some 2" PVC caps to keep the dirt from going up the metal tubing.  It all came to about $50. Then I screwed the augers into the ground, drilled a small hole in the tubing, attached the turnbuckle from the swing tubing to the tie-down with the S-hook on the 4 outside corners of the swing.  It did very well with my now 10 year-old and her friends, and is now doing a great job with my now 4 year-old.  We've even moved the swing set once.  It's a fair amount of work screwing the tie-downs into the spot right under each leg, but it sure works great!  The turnbuckles need to be checked and tightened from time to time.  Over time, the swing set, even with the PVC caps, pushed its way further into the ground, so I ended up needing to drill a couple more holes up each leg to keep the turnbuckles tight.
I've seen people use the type of auger that gets screwed in to keep a pet on a tether, and because they're made with fairly small guage steel, they pull up.  These tie downs have a flat surface that gets screwed into the ground which is next to impossible to pull directly up.  I googled mobile home auger anchors and got hardware.hardwarestore.com  They look like what I got.
Best wishes!

Get some coffee cans and place the legs in them than put cement in the cans then dig holes in the ground and place each can w/leg in the ground a bout one foot deep OR GET SOME TIE DOWN STRAPS AND ANCHOR TO THE GROUND W/STAKES

What kind of swing set do you want to anchor? If it's a standard swing with a swing frame, you need to drill holes in the ground like described in this tutorial.

If your swing set is just a simple one (like a toddler swing), check this how to build a swing set tutorial for an image how to anchor it to the tree.

I had a similar issue with our swing set. There are two things I did to fix the problem: 1) I used larger stakes, and 2) when pounding in the stakes, I angled them toward each other some.
For larger stakes, I went to the hardware store and bought 3-foot pieces of 3/4" galvanized pipe. I have a lot of rocks in my soil, so I pounded one end of the pipe flat to make a sharper point. I had to drive them in with a large hammer. When I pounded them in, I angled the stakes so they were not going straight down, but rather were pointing toward the centreline of the swing set by about 30 degrees. The reason for this is as follows: If you imagine the way the leg of the swing set wants to pull up, if the stake is directly in-line with that,
it can easily move just by getting a little loose. If the ground dries out, or the swingset wobbles a little and loosens the stake, it can pull up. If the stake is at a faily sharp angle from that, it will tend to pull sideways on the stake. That way the stake still offers resistance, even when getting a little loose.

Because the ends of the pipe were kind of sharp, I pounded them down to within a couple of inches of the ground, and used a hammer to knock the edges over a bit. Once I drilled a couple of holes through and fastened to the swing set leg with screws, I mounded several inches of wood chips over the leg to protect the kids.