OK, I asked if there would be any interest in me creating a list of resources for basic sewing and mending skills for people who would like to keep their clothes usable longer but never learned to sew, and it seems that there's some interest. I'm only covering hand sewing because I'm assuming if you need this post, you probably don't have a sewing machine. I've been sewing since I was a kid, but I'm no professional seamstress, so I'm definitely inviting comments and corrections from anyone else who has advice to add! (Note: I'm linking to products at JoAnn because they have a lot of locations and they ALWAYS have coupons.)
I'm going to start out with basic supplies. You can start out by buying basic kit like this if you want, though all that isn't really necessary, if all you're going to be doing is stuff like sewing on buttons. Good fabric scissors are worth it if you're wanting to get into sewing more (and do NOT use them for cutting paper or my mom will appear in a puff of smoke and smack your hand. Paper dulls fabric scissors.) but if you're just snipping thread and not really cutting fabric out, regular all-purpose scissors are just fine. The kind of scissors you get in a kit like this aren't going to be the best quality anyway. However, if you're dipping your toes in and you don't want to make an investment before you see if you like it, they should be fine. Then there's thread quality. This particular kit is Singer brand, so you would hope the thread would be decent quality, but the thread in little emergency kits is often shit. However--it should work OK regardless, especially if you double your thread. I had a multipack of thread I bought at a dollar store when I was in college and didn't have access to my mom's more well-stocked sewing basket and I used that shit for mending for years and it held up OK. If you wear a rainbow of colors and you're going to need several different colors of thread for your mending, getting a cheap multipack of thread isn't a bad idea, even if the thread quality isn't the greatest, unless you're mending an area that's going to get a lot of wear. Generally though, if I'm sewing on a button or mending a seam that's come undone, it's because it wasn't sewn on well in the first place, not because it's been put under so much pressure. TECHNICALLY it's best to use thread of the same material as the fabric you're sewing, but for most basic mending, it doesn't really matter. It's also useful to have a few pins for holding things in place while hemming and to create space for the shank of a button (but you can use a toothpick or match instead if needed). You may also find the needle threader handy if you're not used to threading needles.
Anyway, if all you're looking for is the absolute basics, here's what I'd recommend: just use whatever scissors you already have, a pack of needles (more on which to choose later), and a spool of all-purpose black thread and all-purpose white thread (or whichever colors you wear a lot, but generally white and black will get you pretty far). There are all sorts of thread to choose from, but all-purpose is what you want for basic mending.
Picking a needle: Here's a needle guide from JoAnn. Here's another one. Basically, a pack of sharps will probably be just fine for general use. Here's a pack of assorted needles, complete with threader. . Here's another one. This type of pack of needles is nice because it has a variety of sizes so you can pick something small for hemming something delicate, and a longer needle for sewing a button back on denim.
OK, on to some basic skills!
Tying a knot to get started: This article has both pictures and video for 3 ways to tie a knot. The first one is the way my mom taught me and the way I've always used, but the 3rd one is kinda genius and I might give it a try. The only flaw I see is that even though you're sewing with doubled thread for strength (which I recommend), the knot is basically just a single thread held against the doubled thread, which might not be as strong as actually knotting it like the first variety. I'll also note that the way my mom taught me to use doubled thread is to pull the thread through the needle, put the two ends together, then knot them together like variety 1. That way, your thread is doubled for your knot all of your working length except for the part that goes through the eye of the needle, and you're going to be cutting that off anyway, so it doesn't need to be strong. Anyway, that's my 2 cents. If you want to just use one thread, you might want to consider coating it with beeswax for more strength.
Tying a knot to finish: This is how my mom taught me, but there's no talking to explain what they're doing so idk how clear it is if you don't already know what you're looking at. Here's a good overview of several methods, including the suggestion to "bury" your thread end after knotting, which I recommend. The video at the end shows her finishing several types of projects, so you can see how the different methods work for different situations.
Sewing on a button: Here's a good tutorial on how to sew on a button with a shank, a 2-hole button, and a 4-hole button. Just a note that the 4-hole button part shows how to use a toothpick to give yourself room to make a thread shank so there's room for the fabric around the buttonhole between the button and the fabric it's attached to, but for some reason they didn't do the same thing for the 2-hole button??? I guess that's OK if the button is purely decorative, but if you're sewing on a 2-hole button and you plan on actually using it, use the same method as the 4-hole button to make a thread shank. Note #2: when my mom taught me, she just crisscrossed pins on top of the button and sewed over the top of them to make the space for the shank. If you don't have a toothpick or pins, a match would do, or something else small like that.
Sewing a backstitch: This is a basic, strong stitch that is a good place to start on hand sewing. You want to pull the thread nice and taught so you don't get gaps in your seam when you pull the two pieces of fabric apart, but not so tight that it starts to pucker. This example uses large stiches to make them more visible in the video, but you're going to want to make them smaller so the fabric holds together better. If you're a sicko like me, you'll get a real pleasure out of making as tiny and uniform stiches as possible. Also here's a video of Bernadette Banner showing you how to backstitch, because I love her. However, she's focused on historical methods and a total perfectionist, so probably not the best model for simple mending. But remember that time she made a witch hat, to wear just because? And the Worth-inspired gown? And the endless search for just the right lace insert for her petticoat? Good times. Anyway . . .
Hemming pants: This is how you hem dress pants (if you're hemming jeans, just use a backstitch.) You'll want to just use one thread for this instead of doubling your thread because the idea is to make the stiches as invisible as possible. If you don't have a buddy to help you pin the right length, it is going to be essential for you to check and double check the length because bending over to pin it on yourself is going to mess up the length. It is doable though, it just takes time.
OK, that's all the basic stuff I can think of at this point. Like I said, if other experienced sewers have things to add, please by all means do so! And if there's something you'd like to know how to do that I didn't mention (or if something doesn't make sense) please ask and I'll do my best. Happy sewing! Here's to making your wardrobe last (and hopefully have some fun while doing it, or at least get a feeing of accomplishment.