You can spin yarn on a $15-$30 tool called a drop spindle. Expect to pay about $8 and up for four ounces of soft ready-to-spin wool, which is enough to make a hat. Wool is easiest for beginner spinners.
Ways to Learn
There are three ways I know of to learn handspinning: take a class, teach yourself, or join a guild. You can do one or all.
Check for beginner classes at municipal parks and recreation programs, universities, and independent yarn shops or handspinning supply stores. Classes sometimes provide spinning wheels to learn on. A class will train you thoroughly in the basics. Be aware that after a class is over, the support and structure will be done and you will be left to carry on handspinning independently.
Teach yourself. Start with a one ounce top whorl drop spindle and soft prepared wool top or wool roving from a breed like Romney, Blue Face Leicester, or Falkland. Maybe Targhee. Merino in a pinch. These are for sale from handspinning supply shops (local or online) and independent producers, as well as at fiber festivals. Then search online for handspinning how-to videos; Abby Franquemont (afranquemont on YouTube) has a clear, thorough two-part introduction. Check your library or bookseller for how-to books. Get live advice on the Ravelry website. This method is inexpensive. It requires self-motivation. Expect to be terrible at spinning at first. Slubs are normal.
Pay your dues and join a guild. Ask in advance if beginners are welcome and what, if anything, the guild offers beginners in the way of instruction and guidance. Fiber art guilds usually have a mandate of fostering new interest in handspinning. If you are fortunate, you will find a few guild members who relish the chance to pass on their skills and knowledge on a volunteer basis during the meeting. Be considerate of their time, provide your own materials, and be prepared to do a lot of independent practice in order to get good at spinning yarn. The guild may also offer workshops and classes on a fee basis. Locate guilds through online directories or through staff at supply shops and living history museums. It is common for handweavers and handspinners to be in the same guild; some guilds are one or the other. Regular, ongoing participation in guild activities exposes you to many models and mentors, and increases your likelihood of sticking with handspinning long enough to produce good yarn.
Supplement your knowledge. Living history museums have special events where the public can try handspinning or see it done. I've found staff can be very informative about the properties of fibers, historically accurate techniques, and many other topics. Fiber festivals let you try out equipment and track down specialty fiber suppliers. Even if you can't travel to a fiber festival, look online at the festival's list of vendors. Fiber festivals often have educational displays and demonstrations. Some re-enactment societies have dedicated study groups that focus on textiles; you might find handspinners willing to teach there.
You might associate handspinning with spinning wheels. A drop spindle has a lower price, no moving parts to break, smaller size for portability, the versatility to spin very thick and very thin yarns, silent operation, comparatively efficient use of wood, and a connection to thousands of years of handspinning before the spinning wheel. You can see a short video of me spinning with a drop spindle here.
On the other hand, not all handspinners like using drop spindles, and spinning wheels produce more yarn in less time. A good spinning wheel runs about $400-$600. There are some models that are less expensive. Start with a versatile wheel, not one that specializes in very thick or very thin yarn. Some wheels fold up for easy transport, which is a plus for going to classes and meetings. Try before you buy at classes, supply shops, festivals, and guilds. Do not buy a used or antique wheel unless you are certain that it works or that you can restore it. You will want to have three to four bobbins that fit the wheel; with a used model that's still in production like the classic Ashford Traditional, you should be able to buy new bobbins separately.
As a beginner you don't need toys like wool cards or wool combs because you're buying fiber that is commercially prepared and ready to spin. You might want a lazy Kate for plying, a niddy noddy for wrapping skeins, and a swift for holding skeins while winding yarn into balls, but there are inexpensive workaround substitutes for all these which you can discover with a bit of digging. The substitutes will do for a while as you practice the basics. A well-made drop spindle has a custom hook and a hardwood shaft, a cheap one is made with a cup hook and a piece of dowel. Both work.
Spun yarn needs to be plied to balance the yarn, so whatever way you learn to spin make sure you get a lesson in how to ply too.
You may be interested in my posts, "How to Find a Fleece" and "Reasons for Owning Wool Combs and Hand Cards (Or Not)."
You will get slubs and those gut-wrenching moments where a whole lot of fibre gets caught in the twist and won't draft properly. My advice is to just keep spinning, ignoring the slubs. The twist that gets away on you and runs into the unspun fibre? That twist will lock fibre firmly into a slub. Until you get past that point in your strand of yarn, the twist won't let up enough for you to be able to fix it. Don't overthink it and forget about fixing it. Just keep going. It's like practicing a piece of music for piano lessons: you don't want to keep fixating on the musical phrase you repeatedly get wrong. Think of it this way, the twist is holding on the the fibre like someone's hand tightly grips a cluster of drinking straws. No matter how you tug, the ends are caught. And it's worse because the fibre (if it's wool) is not slick like a straw, it has a surface that catches and locks up. Keep going, keep drafting and spinning, and eventually you will gain better control over where the twist goes. Your fingers will be able to draft the amount you want, not a large slub.
Handspinning is about putting twist into fibre. You'll need to practice putting in the correct amount of twist. Beginners usually overtwist when using drop spindles for two reasons. They want to make sure the fibre holds together, and they are parking and drafting so they carefully work on a short section at a time, putting in more twist than needed. How can you can tell if your yarn is overtwisted? You have a thick strand but it feels very hard and dense, and the strand kinks back on itself a lot when you relax your tension, when you move the working end close to the spindle rather than holding it out.
There is a way you can test and monitor your twist. You effectively preview the type of yarn you're going to get when you're done. Take a section of your strand, about 8 inches, pull at the mid-point, and let the strand fold in half. The halves will twist around each other and look like plied yarn. To undo overtwisted yarn, hold the top of your strand and let the spindle hang free for a moment or two. The spindle will turn backward, undoing some of the twist. Test again and repeat until the little section loses the tight, overtwisted look.
Still, I recommend that you just go ahead and spin. Don't worry about getting your degree of twist correct at the start. It's more important that you get comfortable enough with the spindle that you get past the park and draft stage. When you are able to spin with a drop spindle that hangs and turns freely, you'll find that the spindle will wobble to a stop. Twirl the spindle with a consistent amount of force each time, draft at a consistent rate, let the spindle wobble to a stop, and wind on (or spin and draft again). I find that this gives me the correct amount of twist, no matter how thick I draft or how thin I draft the strand. The spindle naturally spins for a longer time before wobbling to a stop when I draft a thin strand. The only way I can overtwist is if I spin the spindle after it wobbles and I do not draft any more fibre.
If you cannot or will not use animal products but you want to spin yarn, you can find for sale pure natural plant fibres for handspinning like flax and cotton, or synthetically produced fibres made with materials like bamboo or soy or nylon. There are alternatives to dyestuffs of animal origin as well.
It is possible to spin certain other fibres which are gathered from nature and hardly ever sold commercially. Just remember there are many reasons why wool, silk, flax, cotton have won out and dominated textiles for thousands of years. The more you go for offbeat materials, the less likely it is that you will find people who know how to help you learn to spin them, and the less likely you are to get usable cloth for a reasonable amount of time invested.
One other consideration is that plant fibres have a different structure than wool and cashmere; they are more suited to woven cloth than they are to knitted cloth.
Cotton needs a lot of twist to hold together. Try a charkha or a great wheel, spinning wheels that have very high drive ratios. You can try a very light drop spindle or a supported spindle.
Flax needs skill to manage and extra coordination to wet the fibre as you spin.