
Every DIYer's tool box should include a couple of saws. Whether you want to keep it simple with a basic hand saw or feel the power of a buzzing circular saw in your hands, here's a quick little guide to get you started.
HAND SAW

If you're looking for a saw to have on hand for small jobs, a simple hand saw may be enough for you. A hand saw will suffice to chop off table legs, trim the Christmas tree base and other quick DIYs. Warning: cutting with a hand saw is a laborious endeavor -- but it will help you connect with your inner pioneer. Shown here: Stanley 15-inch Hand Saw from Lowe's,
MITER BOX & SAW

If you're going the hand saw route, you'll also find that a miter box, when paired with a back saw, is a great tool to have around. Perfect for cutting trim, molding and frames. You'll want a box that includes clamps or can be screwed to a wooden table. Shown here: Stanley Clamping Miter Box with Saw from Lowe's
JIGSAW

The Jigsaw is a good all purpose saw to have on hand. It's small and apartment friendly (or, as apartment friendly as any saw can be) and can be used for both curved shapes and, with a straight edge, can be used to cut mostly straight lines. Shown here: Skil 5.0A Orbital Jigsaw with Light from Home Depot.
CIRCULAR SAW

For big cutting jobs (think trimming hardwood flooring or cutting big sheets of plywood), you may want to invest in a circular saw. It'll be easier to cut straight lines than with a jigsaw and it'll make short work of the job. Shown here: Dewalt Cordless Circular Saw from Ace Hardware
BAND SAW

Band saws come in two styles: floor models and portable. If you lack a proper workspace, you'll probably want a portable band saw. It can be used to cut intricate shapes in short order. If you do have the space for it, opt for a larger band saw with stand. Shown here: Milwaukee Deep Cut Band Saw AC with Case from Home Depot
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Nomade Express Slee...
Another saw to consider is the Japanese Ryōba saw. This saw has two cutting edges: one for ripping (cuts with the grain of the wood) and crosscutting (across the grain of the wood). They cut on the pull stroke, making them easier to control. There also other single edged Japanese saws too, many with replaceable blades. The big box stores may carry one or two, but stores like Woodcraft or Woodcrafters carry many of them. Well worth the look into.
I have all but resigned my western style saw (the saw at the top) to the back of the shop and almost exclusively use Japanese saws.
I LOVE those Japanese saws. They seriously make it so much easier for someone who is uncoordinated. I haven't ever successfully used a push saw, but man oh man it was easy with a pull saw. The fact that it has a handle like a knife and not a grip like the ones above makes a difference too.
One of the most useful saws, if we're going to include power saws, to have around is a power mitre saw. Some are fairly portable (not much more than a circular saw in weight). Choosing the right blade is key - with the right one you can get very smooth cuts.
As an alternative to the power mitre saw and a bit more precise than the average mitre box, this saw is a pretty good alternative as well:
http://www.leevalley.com/en/Wood/page.aspx?p=32927&cat=1,42884
I would only add to Maybe Someday's post that if you are buying a mitre saw, go for a compound mitre saw. Probably not a lot more $ than regular mitre saw and the bit of extra money is more than worth it.
Japanese saws are the best!
I have a dozuki saw that cuts so smoothly. I never mess with western saws anymore.
http://www.woodcraft.com/product/2000850/2716/shinwa-dozuki-1107029-hand-saw-z.aspx
I have the 15" Stanley FatMax Hand Saw, I don't like it. It is to short, you can not take full stroke, spent the extra few dollars and get the 20" or a 24" saw.
The portable bandsaw is for cutting metal.
A good resource for all things wood online is Rockler. I have actually been able to call them up or email them and get advice on which tools and supplies I need.
http://www.rockler.com
A small thin blade handsaw sold for gardeners will cut small to medium branches in a messy tree.
For potential circular saw users, I find two things that make life easier: a saw cutting guide and 2" foam insulation sheets.
The saw guide lets you cut lengths of plywood safely and accurately. They are very easy to make and magazines like Fine Woodworking, Wood Magazine have online instructions on making them. They are better than the store bought guides since they let you cut on the line without the hassle of offsetting the measurement for the base plate to blade distance.
The foam sheets lets you cut plywood on the ground, instead of balanced on a sawhorse. Lay the sheets on the ground, place your plywood on the sheets, clamp your saw guide on and make your cut. You'll always be working on a stable surface and never be over reaching. Word of advice: adjust the saw so only about 1/8-1/4" of the blade protrudes through the wood.
I hope these tips help.