To get started sewing by hand you will need some needles, fabric, and thread.  For the needles a variety pack will work great.  I wouldn't recommend the packs that come in circle plastic containers, I always have trouble with them.  Fabric can be something that's hems are giving out, a patch to put on your jeans, or a lot of fabric for a project like maybe a quilt.  Next you need thread, and not all thread is created equally.  The 5 for a dollar kind is useless, its so thin it breaks often.  I prefer the fancy thread that's specially for hand sewing, its really think and kind of slick so it goes through the eye of the needle well and doesn't tangle up as much.  Generally if its a dollar or more its decent thread, but in general you get what you pay for.  There are some things that help in sewing by hand that aren't necessary, but very helpful.  If you have problems threading your needles, they make needle threaders.  Also, bees wax helps if you are having trouble with your thread getting tangled.  My grandma always sewed by hand, she never trusted a machine.  She made elaborate quilts that were huge, all by hand.  One of the benefits of sewing by hand is more accuracy.  Not to say sewing machines aren't accurate, but if you do it by hand you can line things up and stick the needle in without moving the fabric.  This is useful when making quilts that are very complex or need things to be lined up.  Its also a lot cheaper and smaller than a sewing machine.  The draw back to sewing by hand is its very time consuming.  Its also isn't as strong of a stitch as a sewing machine.  Eventually you'll do both hand and machine, for things like pillows you must do some by hand to get it to look right.

Buying a sewing machine isn't as easy as you may think.  I had to buy a new one about a year ago because I did something stupid.  I was in a rush to clean up, so I put my sewing machine on the top shelf of the closet.  Well needless to say when I opened the closet the next time my sewing machine fell down.  It was completely broken, the shop said it would be cheaper to replace.  I had just started to get the hang of it too.  I was so upset with myself.  Off I went to get another one when I realized I hadn't a clue what to get.  My first one was a gift, so I didn't know exactly what to look for.  After much research and shopping around, I found a Singer that I liked.  Now most say Singers aren't any good, but that's what my first one was and I knew how to work it.  You can get a decent sewing machine with a couple different stitches for around $100.  The different stitches are for different kinds of fabric and projects.  You probably dont need the sewing machine with 1000 different stitches.  The ones you need are depending on what you plan on sewing.  If you want to make work clothes, then make sure there are stitches for the kinds of fabric you want to use.  If you want to make blankets, then there are blanket stitches.  A basic machine should have some or most of these stitches.  You will have to read the manual to see how to thread it, as all machines are different.  Here are some things to know about machines in general.  Most is self explanatory, but if you're thick like me someone telling you might help out.  Tension is how tight or lose the stitch is, if it's to tight you'll have problems with the thread breaking, if its to lose then your stitch will come apart easily.  Also the tighter or closer you have your stitches as effects too.  The tighter your stitch the strong the hem, the loser the stitch the easier it is to fix mistakes but it can cause the stitch to come apart easily.  If the stitch is lose on one side and not the other that means there is a problem.  If it's on the top it means that there is a problem with your bobbin, if it's on bottom then it's your threading. 

Fabric is obviously the most important part of sewing, without it you would just not have any purpose for the rest of the stuff.  Seriously, where you buy your fabric is important.  Wal-Mart has some really cheap fabrics, JoAnn Fabrics has some that's a little more expensive but a great selection, and then you have quilting fabric stores where you can pay up to $10/yd for cottons.  Wal-Mart and JoAnn's has about the same quality of fabric, so if you can find what you need at Wal-Mart don't bother with JoAnn's.  I do think JoAnn's is a little nicer, but I am a cheapo so I go to Wal-Mart first.  JoAnn's almost always has the fabrics I want, so if you want something you know is a little out of the ordinary go to JoAnns.  Quilt stores has quality fabric (thick and high thread counts), so if you want something that lasts several generations, you may want to consider forking over all the money for nicer fabric, otherwise its just a waste of money.  Most people pre-wash their fabric so that it doesn't shrink and change sizes.  I found most patterns have been adjusted to this and are big enough to be able to take a shrinking.  For quilts I like it when it bundles up a little cause of shrinking, but that's just me.  Maybe I am just too lazy to pre-wash, but that is something you need to think about when sewing.  Regardless, always wash things alone for the first time so you don't get it bleeding through on something. 

Simple projects are things like pillows, curtains, and table runners are really simple to make.  Here are the basics on how to make them.

Pillows-

Cut out two squares the same size.  Sew 3 sides together, turn it inside out, and add stuffing.  When adding batting (stuffing) break the stuffing down as much as you can, don't ball it up.  Batting has a memory, so if you ball it up, then you'll have a big ball in your pillow.  If you break it down, its easier to put it in the corners and to fill them better.  When you have it all stuffed, you will need to stitch the open side up by hand.  Put your first knot on the inside where the stuffing is to hide it some.  Then stitch it up with a loop stitch, check out your couch cushions to see how they are stitched if you don't know how to loop stitch.  When you get done tie a knot at the end as close to the fabric as you can.  Then poke the needle some place close to the knot, and wiggle the needle around so it makes a bit of a whole.  Then put the needle through the hole and have it come out somewhere far away from the hole.  Now grab the fabric near the hole and tug on the thread and needle end.  If you do it right the knot will slip into the hole.  This is a way of hiding your knot.  Now pull on the needle and trim the thread as close to the fabric as you can.  When you wash it the hole will go away and you don't see either knot.

Curtains-

Take a big enough piece of fabric to cover your windows and then some.  Cut it in half where you want the split.  Sew a hem on the sides and bottom.  For the top you can either sew it like a hem but with enough room to put a rod through, or you can hem it and sew hemmed strips of fabric or ribbon works too and make loops with them to go over the rod.

Table runners-

Table runners are a great way to decorate for holidays.   Get some fabric of your choice, the top and bottom can be different colors if you'd like.  Draw out the length and width of the table runner you'd like.  Cut out both fabrics and some batting (thin type for quilts works best) to the same size as the pattern.  Curl one of the fabrics around the batting and pin it down.  Curl the ends of the other side so its got the hemmed look and pin it to the batting and other fabric.  You can remove the first set of pins if you'd like.  Now just sew a border around them.  Make sure there is enough fabric going through all layers, you don't want to have a hole.  It may help to cut the corners at an angle to fold them easier.  You can add a tassel or something at the end if you'd like.