The clarinet: Yamaha has been a musical instrument manfacturer for over 100 years (starting with organs and pianos). Yamaha plastic student instruments are fantastic, and the quality control at Yamaha is excellent. They are expensive compaired to the clarinets that you might buy at Target, but then, Yamaha clarinets will be playable. Btw, Yamaha plasic clarinets will hold up to the abuse of younger students and play in tune. I own only two Bb soprano clarinets, and one is a plastic Yamaha. (They come with a decent mouthpiece as well, so you can wait on that purchase at least a year or so).
The Buffet B12 is a nice plastic clarinet - and has reliable tuning and keywork.
There are other trustworthy brands, Selmer-Conn/Leblanc/Vito (all one company at the moment). The Vito clarinet was one of the most popular student clarinets that my company sold, but quality control issues make it difficult to recommend unless it was checked by a qualified professional first. The author owns a Vito Bb Bass clarinet. Avoid clarinets sold by the big box retailers, Target & Walmart (unless you want to be discouraged and miserable when playing the clarinet) They have atrocious feel, and owing to the price, likely have miserable durability in the long run. Contrary to popular wisdom, you want your clarinet to last, because its better to spend $800 for a clarinet that you like now than spend $150, and then $800 for the clarinet you should have purchased in the first place. If you are a concerned parent that fears that the $800 will be waisted, then rent an instrument at first, or buy a slightly used Yamaha clarinet.
Reeds are simple to choose when you first start playing clarinet - get ones soft enough that you can practice a correct embouchure and breath support. Synthetic reeds are more expensive per reed, but they are easy to play and last longer than natural cane reeds. Don't throw away reeds that don't play for you right away - with practice (or a change in weather)those reeds may become playable later.
Mouthpieces are the most important part of the clarinet (besides the player) for tone production. The Pyne Polycrystal mouthpiece, which has great tone production and ease of play, has the added avantage of being durable and inexpensive. The mouthpieces that come with Buffet clarinets are horrible, but the Yamaha mouthpiece is playable. Other mouthpieces might be recommended by you teacher or private instructor, but there is a caveate - hard rubber mouthpieces are as fragile as glass. They are easily damaged by dropping. And the metal caps that are meant to protect the reed can just as easily badly damage the tip of a hard rubber mouthpiece.
Ligatures don't matter too much at this point. The ligature should be good enough to hold the reed securely, and shouldn't damage the mouthpiece. The Rovner fabric ligature does both quite well, but the Bonade regular ligature (screws on the reed side) is my preference. Both hold the reed in place, and are easy to adjust.
Miscellanious equipment that a beginner needs is cork grease and a swab - silk doesn't get caught in the instrument under any circumstances.
Notice what is not mentioned here: Keyoil - hardly ever needed, and then not something an amatuer need worry about. Wooden Clarinets - band students (at least eventually) need a plastic clarinet for marching in the bad weather. A wooden clarinet is prone to cracking and needs special care; skip the hassle with a plastic clarinet. Expensive reeds provide little or no benefit at this point; just getting used to clarinet will require softer reeds, and tone production is not important until you've developed the mucles to play properly. Tuning Rings - not yet useful to beginners. Special Barrels - If you are thinking about a new barrel, are you sure you are a beginner?