My philosophy is the same as any other genealogy problem: start with what we know and work toward what we don't know. DNA is just another tool in our genealogy toolkit.
Upload Your Tree
First, take the time to put your ancestors into an Ancestry tree linked to your DNA results. Make it private or make it public -- just get it done. In the background, your tree will be matched to the trees of your DNA matches and in time (a lot of time) you will get common ancestor hints. I removed my tree in early May after about 70 days online and replaced it on May 8th. As of now, May 27th, there are still common ancestor hints that have not reappeared.
I also suggest, if your tree is private, that you add any theoretical people so that you might get a match on them. For example, I have a possibility of an ancestor named Abednego Carter. He's now in my private tree, not publicized to anyone, but just hanging out looking for a common ancestor match.
Look for Hints
Check every few days for the common ancestor hints. This is not the same as the "shaky leaf". The hint in the DNA matches is that you have a common ancestor. One day there are two hints and the next day there are three. You won't be notified -- you have to look for them yourself. Use the filter "Has a Hint" and slide the relationship range to both ends. These hints are not perfect. I have one such hint that is actually wrong because the other person has some incomplete information.
Use the Optional Notes
See the little note symbol above with each match? Jot a note in there to remind yourself of what you noticed. It pops up when you hover over it with your mouse.
Starting with your closest cousins and those with shared ancestors, see what they know that you don't. Ask about their sources. If you have information they don't have, share it with them.
One of my 7th cousins had a woman's surname that I don't have. She didn't have a good source, but she gave me a clue to work with. In another line, I sent a copy of a probate packet to a 6th cousin who hadn't seen it.
See my 2nd cousin at the top of my match list? She and I compared notes on our 3rd cousin matches and found that one of them is in both our match lists. Now we know which of our lines he relates to. We can work with him to identify another generation, most likely in his tree, possibly in ours. Helping him may help us and he has the potential to become another collaborator in our shared line.
Work Near to Far
Work those 2nd and 3rd cousin matches before moving on to the 4th and 5th. If you don't know how you relate to the 2nd or 3rd cousin matches, figure that out. The more you know about generations close in time, the easier it is to work back in time.
Stay Alert, Reach Out, Share
I was probing a 3rd cousin's information ("Bud") and found that I kept looking at "Tim's" tree for hints, as it was far more complete than Bud's. Sure enough, Tim is in my 5th cousin matches.
It turns out that Tim is the nephew of Bud, which then would indicate that 3rd cousin is not accurate! It is more likely that Bud is a 4th or 5th cousin. So I've mentally set him aside for now to work other 3rd cousin matches.
I also sent Tim a message about a family book which will help him with his research, as his known ancestors are intermarried with mine. He was excited to see his ancestors named, including two generations he didn't yet have. We don't have a common ancestor yet. But now I am watching for his ancestor's surnames, as one of them may be my ancestor. And he will be watching for my ancestor's surnames, as the opposite may be true.
I started by asking how we can benefit from this DNA adventure. I think the answer is in the sharing and collaboration.
I'm reminded of a collaboration from years ago. You all know those lovely pre-1850 census pages -- the ones that are just a series of numbers. I'd been collecting them for a puzzling branch of the family. A county-level researcher asked if I could identify a couple of women for whom he had a maiden name, but not parents. I was able to place them in the family based on a pre-1850 census. Without him, I never would have found names to match the numbers. Without me, he would not have identified parents. We had very different parts of the puzzle, but together we solved it.
I believe that when we help others, others will help us. Those of us who are further along will help those coming behind. Together, we'll all move forward.