Subtitle: it won't get you a show on TLC but it might get you some free soap.
This applies to the US; I have no idea if couponing works the same anyplace else (sorry)!
So let's start with the main way you get free (or almost free) stuff: a concept called "stacking."
Stacking coupons simply means that you have used different types of coupons on the same item. Most stores will allow both a manufacturer coupon and a store coupon to be used on the same item (you can see which kind it is in the box usually near the top, which also says the expiration date). Target, specifically, allows a manufacturer coupon + a store coupon + a Target Cartwheel % off deal on the same item. If you hit that good timing combo (and maybe even coinciding with a sale!), you can often get things for free, for close to free, or even free + money back. Yep - that's called a "moneymaker." It happens when you have, say, $1.50 off in coupons but the thing costs $1.30. Some stores won't give you the change, but some will. I will take my free soap AND twenty cents, thanks!
Pay attention when downloading coupons to store loyalty cards: a lot of them are manufacturer coupons now (even on Cartwheel), when you might presume they are store coupons. ALSO, the card will hit first. What that means, is if you clipped a manufacturer coupon from the Sunday paper for a dollar off, but had already loaded a 50 cent coupon onto your Kroger card, so you want to use the clipped coupon instead, scan your card LAST because it will take that coupon first, making you unable to use the clipped coupon. Similarly, if you use Target Cartwheel (available on smartphones or website), scan that code last as well. Paper coupons first!!
There are lots of rebate apps, which I cannot use because I don't have a smartphone (😢), such as Checkout 51. Check those out because they often have amazeballs deals!
You can find lots of printable coupons (as well as breakdowns of the best deals to be had that week/at which stores) on couponing sites; my favorites are passionforsavings.com and freestufffinder.com.
Some Helpful Abbreviations When Browsing Couponing Sites:
RP = Red Plum, one of the coupon inserts in the newspaper
SS = Smart Source (insert)
P&G = Proctor & Gamble (insert that comes out once a month)
Where to get coupons without a newspaper subscription:
First, if you know anyone who still gets a paper, you can simply ask them for any coupons they don't use. You might also be able to get just the coupons part of the newspaper delivered for free. I say "might," because the Denver Post says this is a thing, so I signed up for it, got them for about a month, then since then I only get them on Big Shopping Holidays (Black Friday, New Year, Memorial Day).
You might also be able to get coupons from a café: my local Starbucks has a newspaper rack for people to leave their paper they bought and read over coffee in case anybody else wants to read it without buying a new one, as a sharing thing. One day I was in there and the Sunday coupons were in the share rack, so I asked the barista if I could abscond with them, and she said sure, because otherwise they'd just be recycled. There isn't always a paper in there, but it's worth a try!
Organizing tip: use a file folder type system to organize your coupons. Keep them in their intact inserts from the paper, because they will have the date printed on the spine. When you go to a couponing site, and they say the coupon is in the 11/5 SS, you can pull it out right away rather than rifling through stacks of coupons:
Never walk into a JoAnn Fabrics or Michael's Crafts without a coupon. They send coupons to my email every day it seems. Sign up if you're a crafter (or use the one in the paper, if you get a newspaper).
Petco, as far as pet supplies go, sends the best coupons, IMHO.
King Soopers/Kroger has brought back double coupons after a hiatus!
Some things I have gotten for free/near free/moneymaker? Soap, razors, cat food, dog food. No kidding!
There ya go - couponing for things you actually need, 101. I hope it helps you stretch some dollars!