Instrument
It will come as no surprise that a student will need a good quality instrument to start. Please avoid the E-bay and Amazon $99 instrument like objects. Some of these may play fine at first but will soon wear down and become unplayable and unrepairable due to the materials used. Your best bet is to get an instrument from a reputable local music store.
Swab
A swab is used to remove the moisture from the instrument after playing to keep it in the best possible shape. I generally recommend silk swabs with rubber coated weights for beginners. These rarely if ever get stuck inside an instrument and the rubber coated weight allows you to use it on your mouthpiece without fear of damage. You may need to get a different swab for the neck and mouthpiece of the saxophone. Most music stores have reasonably priced swabs of this type from a variety of manufacturers.
Cork Grease
Any brand of cork grease is great. It is used on the corks of the instrument in order to make the instrument easier to put together.
Pencil
You ALWAYS need to have a pencil with you at all rehearsals and practice seasons. They are the only acceptable way to mark music since it is erasable. I recommend you use just the cheap mechanical pencils available so you do not have to worry about sharpening them and it will not be an issue if your stand mate “borrows” it for an extended amount of time.
Reeds
Reeds are some of the most important pieces of equipment in a reed player's case. It is a rare moment when a group of musicians who use them get together and the subject of reeds does not come up. There are more opinions on reeds and how to deal with them than musicians.
For beginners I recommend either Vandoren Juno reeds or D'addario Rico, Royal, or La Voz. It may come as a shock that Rico, La Voz, and Royal are the same reed. The only differences are the markings and that the Royal reeds have a french file. Use what is available, inexpensive and comfortable. For reed strength beginners should start with either a 2 (soft), or a 2.5 (med-soft) in the previously mentioned reeds.
There are other “premium” reeds that many musicians use and recommend. They tend to be much more resistant to play on and difficult for beginners to use. When you are ready to try different reeds you should seek the advice of a teacher or mentor.
All clarinet and saxophone students should have a minimum of 3 to 4 playable reeds in their case at all times and rotate between them. I recommend that students use a basic 4 reed reed guard to help keep track of their reeds and or use a sharpie to mark them. Use the following plan to make sure you always have good reeds.
Week 1 - Start 2-3 reeds and alternate them with every practice session or day.
Week 2 - Add 1 reed to your rotation.
Week 3 and beyond - Add 1 reed to your rotation, and toss out the worst reed if your reed case is full.
Please note that this is just the start of your knowledge of reeds. I am in fact still learning about them after over thirty years of working with them.
Neck Strap
A neck strap is essential to properly holding the saxophone and can make playing the clarinet easier on the hands. When looking for a neck strap one should look for one that is easy to adjust but will not slip, has a good instrument hook that will not fall off the instrument unintentionally, and is comfortable. A great example of a strap that follows all of these ideals are the ones made by D’addario (Saxophone & Clarinet Neck Straps | Accessories | D'Addario (daddario.com)).
Music Stand
All students need a music stand in their practice space. Any good quality music stand is acceptable. The standard is still the model 48 stand made by Manhasset. These have been in almost every band room and concert hall I have visited. This is a buy once, cry once item (I have had one of my Manhasset stands for over 20 years and I received it used so it is much older). You cannot go wrong with either the model 48 (https://www.sweetwater.com/store/detail/M48--manhasset-symphony-stand-single) or the model 52 (https://www.sweetwater.com/store/detail/M52--manhasset-voyager), which is a folding model.
Tuner and Metronome
Students will also need a tuner and metronome. There are numerous ones that you can get free as apps on your phone or computer. Those are great to have but I tend to prefer to have a separate device on my stand (yes you may blame my age). A good all in one solution is the Korg TM-60 (https://www.sweetwater.com/store/detail/TM60BK--korg-tm-60-tuner-metronome-black). If you choose to use an app on your phone I recommend the app Tonal Energy (https://www.tonalenergy.com).
Chair
One of the most overlooked items that a musician needs in their practice area is a good quality chair that is stable and of an appropriate size for the student. The chair should not have arm rests as these will get in the way.