First, if you spend any time at a music store, you will notice many brands and styles of reeds. Let's talk about some options. It is cheaper to buy reeds in a box rather than individually. Always buy reeds for your type of saxophone i.e. alto buys alto reeds, tenor buys tenor reeds etc. You will also see that reeds are sized in numbers 1-5, and in another system that goes soft to hard. Most reeds are using the number system. More expensive does not mean better. I would start out with a Rico Royal size 2.5. If you go with a Vandoren reed (a little more expensive) go with a size 2. The Vandoren reeds and the Rico Royals come in a variety of different cuts. On Vandorens choose the one in the blue box. They also make Java, V-12 etc. but just choose the regular old blue box reed. On Rico Royals, just pick the Rico Royal brand.
Not the Jazz cut and not Rico. Rico is not the same quality as the Rico Royal. The size of reed you choose has to do with a couple of factors. The biggest factor is probably the size of your mouthpiece. Another factor is embouchure strength. After a couple of months of steady practice, you may want to move up a 1/2 size.
In general reeds need to be soaked before you can play them. A reed can be soaked in a cup of water, or in your mouth. All parts of the reed must be completely soaked before it will play its best; this usually means about two minutes in the mouth. Don't forget to soak the back! Take care that you don't nick the tip of the reed with your teeth. If the tip appears to be "wavy" soak it longer. After three minutes you can set the reed on a flat surface and rub your thumb from the back of the reed to the tip and off the end. NEVER rub the reed from the tip back up, you can damage the tip and make the reed worthless. Once the reed is properly soaked, take your mouthpiece and lick the table. Now take the reed and line up the tip of the mouthpiece with the tip of the reed. I prefer to line it up just a little short of the tip of the mouthpiece. If the reed is over the tip, it won't play the same. You can never be too picky on the placement of your reed. ALWAYS make sure it is perfectly lined up.
Now, take your ligature and slide the big end onto the tip of the mouthpiece. Be careful not to touch the tip of the reed with the ligature. The ligature will always be placed with the screws on the right side when the reed is facing you. Sometimes the screws are on the top and sometimes on the bottom, but always on the right. You'll notice that the ligature is tapered (big and little ends). Your mouthpiece is also tapered. You will need to put the big end on first so it slides down to the big end of the mouthpiece. Put the ligature back far enough so that it is on the bark portion of the reed. Fasten the screw(s) tight enough so that the reed will not move around on the mouthpiece.
A word on reed care. Reeds are expensive and delicate. They will easily break if you touch the tip of the reed with your finger. To move the reed around on your mouthpiece always use the sides and back of the reed to move it and NOT the tip! If you live in a dry climate, your reed could dry out while you are resting for a few minutes. Always keep your reed wet when it is on your mouthpiece. If you are in school and your teacher is talking to another section, you can put your mouthpiece cap on your mouthpiece to keep it wet. A good trick is to lick the reed first, then put the cap on. I prefer taking some tape and covering the holes on the mouthpiece cap. If you live in a humid climate, you won't have as many problems with reeds drying out. Keep in mind that air conditioners and heaters take the moisture out of the air.
Another useful tip to help reeds last longer is to rub the reed with your thumb from back to front pushing very hard. This should be done on a flat surface. This will crush the fibers in the reed and make the reed "waterproof". When you are done playing you should always remove your reed and place it in a reed guard. Several companies make reed guards. I'd probably get one that holds four reeds. You should always have four good reeds that you rotate on a daily basis. When one is no longer usable, replace with a new one.
Wipe off your reed before you place it in the reed guard. The key to keeping a reed from warping is to let it dry out slowly. If a reed dries out quickly, the thin ends dry first and the reed warps up. When it is warped, it will no longer lay flat on the table of the mouthpiece, and will need to be sanded down. This subject is a little complicated, and I won't address it in this article. You can prevent reeds from drying quickly by putting them in the reed guard, placing the guard in a plastic bag and folding it once. You don't want it to be air-tight. If it is air-tight, you will grow mold on your reeds. Wash it off, and don't fold so tightly next time. If mold continues to be a problem, take a teaspoon of rock salt and wrap it in some old pantyhose. This will keep the mold away.
The average reed will last about 20 to 30 playing hours. If you break the tip or take a good chunk out of it, it may last only 30 seconds! You will know when a reed is no longer usable, because it will be broken at the tip, or the reed will feel soft when you play. It will blow really easy, and close off when you play higher. If a reed is not sounding good, and it sounds like there is too much air in your sound, the reed may be too hard, or may not be on straight. If you do play on a reed too long, it can become water-logged. It will appear to have water inside of the reed and will look darker in certain spots. You want to avoid this by switching reeds daily. TAKE CARE OF YOU REEDS AND THEY WILL TAKE CARE OF YOU!
A note on synthetic reeds. I have played a lot of reeds in my day, and discovered that the plastic reeds generally don't sound as good. The only exception to that is the Fibracell reed. I love these reeds. They sound excellent and can be played for up to a year. They will break, but are water proof, always flat on the back and will not warp. I'd start out with a cane reed, and later if you want, try a fibracell with a similar size to the cane reed that you are playing. Just recently I've had good success with a brand called HAHN. It might be worth checking out!