Humans are not the only creatures that populate the Disc. Trolls, Dwarves, Imps, Gods, and others interact as people, but the other creatures of the Disc are no less important. In fact, for a witch, they are far more important.
Some creatures we can call to our aid, like fireflies and fruitbats, some we need help from for spells and teas, and some we keep with us as familiars just for fun (and character). Stuffed animals are another cute! option, but I won't go into that here. There are also creatures which a dangerous that roam the disc, but that's another story. Those we don't want following us or hiding in our pockets.
Creatures that can be summoned:
Creatures that are useful:
Creatures that can be familiars:
Fireflies can be summoned with Hag's Blessing. They like jam, come in swarms, and provide light as long as they stick around. You can have up to ten swarms around you at any given time; more than that and they get competitive. Don't attack your fireflies, ever; they are VERY dangerous when provoked. NPCs who are stupid enough to attack them apparently are quite entertaining. They can only be summoned outside, but will follow you anywhere until their time is up.
Fruitbats can be summoned with Mother Harblist's Fruity Flyer. They like fruit, you can only have one at a time, and they are extraordinarily useful for carrying stuff for you. Plus, they're cute. Fruitbats come in a random sex, but you get to give them a name. Once you have a bat, try syntax ask to learn all the things you can ask it to do, which will include returning items to you, staying, and following again. They can carry a surprising amount, including money, but it is possible for them to lose things--and items lost on a bat will not be replaced. Bats require constant feeding, though they will let you know when they are hungry. Having them lug around a stock of dried or pickled food (melons are popular and, when in season, can be acquired for free in the melon fields, but apples and other fruits are good as well). You will want to acquire a whistle at Harmony Stardancer's and bond it to the bat. That way, if you lose her, you can blow the whistle and she will try to come to you. If you leave her outside, by asking her to stay, you can fly to her (since she has a name) on your broomstick. Bats can be attacked, injured and killed by certain NPCs and enviromental hazards. They will also leave if you do not keep them fed.
Bees are a specialty of Granny Weatherwax's. She has even managed to "borrow" a hive, which is quite an achievement. She keeps several hives behind her house for honey and company. We witches can use the bees (and their honey) in several ways.
Bees can be summoned with Mother Feelbright's Busy Bees. This spell uses a glob of honey to attract a swarm of attack bees, which will sting and weaken (or even kill) the target. Victims will attempt to run to escape them.
Honey can be acquired by finding a hive, getting a honeycomb from it, and shaking out the honeycomb. Honey can be used to summon bees with Mother Feelbright's Busy Bees or make teas like Cold Cure. You can increase the productivity of the bees, so that you can get honey faster, by using Biddy Amble's Bee Buzzer.
Frogs are the witchy little creatures you can use to preserve fruits and herbs using Mother Twinter's Potent Preserver. Everything you need to get a frog, including the frogs themselves, is in Granny's cottage. Look around carefully. Frogs will get frustrated and hop away after too many drying-outs.
Goats are useful creatures in two ways: They like undergrowth, and they produce goat pellets. To make smelly seeds, you'll need both abilities. To get ivy seeds, you'll have to get on top of Granny's house, and the ladder tends to get buried in undergrowth. Luckily, goats eat undergrowth...and will follow it, too. As for pellets, if you go into the goats' lean-to at Granny's, you can get and enchant as many pellets as you need.
Ducks are useful primarily for their feathers, which you use for the traditional casting of Granny Lipintense's Layer of Lard. There is at least one pond, I believe in Hillshire, with a sign that says "Do not chase the ducks." Well, you're a witch; rules don't always apply to you, now do they? I expect they just forgot to write underneath "Except if you're a witch, then go right ahead."
Owls are also primarily useful for their feathers. There's one in Granny's Lean-To that never quite seems to leave. You need the feather to make Intelligence Tea.
Spiders are one of the components to Anti-Poison Tea. They can be caught in Granny's chimney. Sadly, they're tricky to catch and it's usually fatal for the spider, but that means you don't have to feel as bad for them when you grind them up later for the tea.
Eels are useful for their slimy skin, which you use in Granny Lipintense's Layer of Lard. You need a knife or dagger to get the skin off the eel. They can be found, ironically enough, at the same pond where you can find the ducks.
While any citizen of the Disc can get a pet from the pet shop in Ankh Morpork, they only sell the Black Cats to witches. Everyone else gets Tabby. You get to choose the sex and name of your pet; I have a male named Mintaka. Cats are fun; they will sit, stay, and meow happily. They will willingly follow you anywhere (as long as you have their licence on your person), and follow you again when you call their names. The rub is that they weigh five pounds and must be carried or left behind when you fly. For a weak witch, this can sometimes be problematic. But a black cat is about the witchiest familiar you can have, and they are so cute!
At the Sunshine Sanctuary, you can get a Swamp Dragon. They make lovely, adorable pets (and like to eat coal, I believe) but I've heard they are troublesome and explode rather too easily. They do have a nice variety of colors, but I have yet to own one so I can't give you any more details just now. Time to do some research, huh?
The AM Pet Shop has a lot of options besides cats, if you are interested in a different sort of familiar that more suits your idiom. Dogs, owls, and an interesting collection of breeds are available, under basically the same rules as cats. Once again, you will have to carry them and watch where you take them, since even owls do not (unlike fireflies and bats) follow you in flight.