I have recently increased my commute from 1 mile each way to 16. It's a mix of city riding, country roads(an area I call no man's land where it's a two lane road with a 55mph speed limit, chip sealed road, all kinds of debris, rolling hills, and for some reason always a strong head wind), and some mixed use tails. This week alone I have suffered 3 flats(one slow leak that I couldn't determine the source, one from a piece of a steel radial tire, and one total blowout). I'm running 700x28 tires, but can go up to 32's. I'm thinking of running a multi tiered flat defense, starting with a good puncture resistant tire( Conti Gatorskins or similar), adding a liner, using sealant in my tubes, carrying a flat kit and a spare tube. Is there some piece of the flat puzzle I'm missing? This is getting to be more expensive than driving. Please help.
I think that you should only take such drastic measures if you really required them. What tires are you currently using that are getting so many punctures?
I would try for something like GatorSkins or Schwalbe Marathons by themselves to see if that fixes the issue. Only once you determine that a good tire by itself isn't sufficient should you go further and add tire liners. Sealant should be a last resort just because it's messy to deal with if you get a larger puncture that the sealant can't fill.
Make sure your tires are always filled to the correct pressure in order to prevent pinch flats. A good track pump makes it easy to top up your tires a couple times a week.
Definitely carry a tube or 2. Flats seem to happen in the rain which makes patching harder. The roadside puncture kit should have a sealed tube of glue as it goes off once opened (unless you get on with glueless patches). Once I've opened a tube I patch any punctured tubes, which become my spares. With old-fashioned patches this works. Then your only consumable is patching supplies (and the rare blowout; I've had one in 25000 miles - overinflation).
I can vouch for marathon plus too, but I'm currently riding marathon supreme which should be a bit faster while still having a decent level of protection.
I run liners on another bike and they've caused one puncture but saved a few. On that bike it's broken glass I'm most worried about. I only use them on that bike because I was too cheap to do it properly. With good anti-puncture tyres, liners would probably do more harm than good.
Slime works well but can be a bit of a pain if you have to let the tube down (e.g. replacing a tyre, rim or spoke).
Suggestion: go for gatorskins or marathon plus/sumpreme. Add slime only if that's not enough. Keep the pressure up.
Have a look at this other question, it has a lot of good information: What is a good way to keep my tires from going flat easily?
FYI, the main prevention for flats is good tire pressure. For 28mm tires try running close to 100 psi. I run 23mm medium weight road tires at 120psi and get one flat a year on similar roads and distances to what you describe.
Technique should not be discounted too - stop riding through roadside detritus and go around it.
Learn to unweight your bike if there's unavoidable potholes, or ideally learn to bunny hop.
Depending on your location you may be entitled to take the road lane (this is also a great speed motivator.)
Another technique to revise is how to fit your tubes/tyres. A mistake in fitting can contribute to later punctures, so never use screwdrivers as levers only plastic tools if you need one at all.
You should also practice putting the tyre's logo at the valve stem. This way when you find the puncture in the tube you can lay it out on the tyre, and identify a 50mm area of the tyre to inspect closely. Makes finding the cause a lot easier.
Punctures are a part of riding sadly, so don't be put off by the odd flat.
32 mm puncture resistant with slime is good prevention.
I don't like a liner as it abrades the tube.
I don't carry slime as the spare as they take up a lot more space.
I took the Schwalbe Marathons off my bike as they are heavy and harsh. But they are about the most ballistic.
Inspect your tires on a regular basis and clear debris! You often have small glass that will work its way into a puncture. Keep the tires at recommended pressure.
In contrast to Criggie's answer I use Marathon Plus (size 700x32) and don't get punctures.
I started using them because I used to get occasional punctures and didn't like that.
I even found they're faster, too, if only because I'm more confident, that potholes and whatnot have no effect on them.
You say you had 3 flats in 1 week (where "1 week" is presumably 160 miles). I have no flats in 1000s and 1000s of km.
I found they have to be newish, after 5 years the tires lost their magic i.e. became vulnerable to slow leaks, partly from wear but maybe because the rubber dries out or something even if they're not too worn (I replaced them, and the new tires are magic again).
If you are getting flats very frequently, you probably have some glass embedded in the tire and when you hit a bump at the right place, it pierces the tube. I had tires that looked like they were in good shape but I was getting flats on every ride. When I got two flats on a 50 mile ride I decided to switch to gatorskins. I then suffered a catastrophic blowout with that tire where the bead separated but the manufacturer replaced it at no cost. I also went to larger tires on the recommendation of the bike shop. According to the dude, the larger tires are actually 'faster' because there is less contact with the road. I'm not sure about this 'fact' but I haven't had a flat since, it's more comfortable and I don't really notice much difference in performance.