So, what are the "proper" plants and animals?
Well, I've done research (built off other people's research, of course), and compiled it here for easy access.
Here's a list of forty animals which form the basis for early-culture domestication. I'll break this into three sections: large domesticatible animals, small domesticatible animals, and semi-domesticatible animals. The semi-domesticatibles are ones where various individual factors (such as ability to breed in captivity, or demeanor) prevent full domestication from occurring. I've also included how each animal can be useful to the culture domesticating them.
Alpaca (Vicugna pacos) USES: fibre, meat, show, pets
Domestic Bactrian camel (Camelus bactrianus) USES: milk, transportation, working, hunting, plowing, draft, mount, fighting, show, racing, meat, hair
Domestic pig (Sus scrofa domesticus) USES: meat, leather, research, show, racing, fighting, truffles, pets
Yak (Bos grunniens) USES: milk, transportation, working, plowing, mount, racing, fighting, meat, fibre
Domestic dromedary camel (Camelus dromedarius) USES: transportation, working, hunting, plowing, draft, mount, show, racing, fighting, milk, meat
Bali cattle (Bos javanicus domestica) USES: meat, milk, show, racing, working, plowing, draft
Donkey (Equus africanus asinus) USES: transportation, working, plowing, draft, mount, meat, milk, pets, racing, guarding
Domestic goat (Capra aegagrus hircus) USES: milk, meat, fibre, skin, show, racing, fighting, clearing land, pets
Horse (Equus ferus caballus) USES: transportation, meat, working, guiding, servicing, hunting, execution, plowing, draft, mount, fighting, show, racing, milk, pets
Water buffalo, including "river buffalo" (Bubalus arnee) and "swamp buffalo" (Bubalus bubalis carabenesis) USES: working, plowing, draft, mount, fighting, meat, show, racing, milk
Zebu (Bos primigenius indicus) USES: meat, milk, leather, hides, working, plowing, draft, vellum, blood, transportation, soil fertilization, fighting, show, racing
Gayal (Bos frontalis) USES: meat
Cattle (Bos primigenius taurus) USES: meat, milk, leather, hides, working, plowing, draft, vellum, blood, transportation, soil fertilization, fighting, show, pets
Llama (Lama glama) USES: transportation, working, draft, pack, meat, show, racing, pets, guarding
Sheep (Ovis aries) USES: fibre, meat, milk, leather, pelt, vellum, pets, show, racing, research, guarding, fighting, ornamental
Domestic goose (Anser anser domesticus) USES: meat, feathers, eggs, show, guarding, pets
Domestic duck (Anas platyrhynchos domesticus) USES: meat, feathers, eggs, pets, show, racing, ornamental
Domestic pigeon (Columba livia domestica) USES: show, ornamental, messenger, meat, racing, pets
Chicken (Gallus gallus domesticus) USES: meat, eggs, feathers, leather, show, racing, ornamental, fighting, pets
Ferret (Mustela putorius furo) USES: pets, hunting, pest control, show, racing
Domestic turkey (Meleagris gallopavo) USES: meat, feathers, eggs, show, pets
Domestic silkmoth (Bombyx mori) USES: silk, animal feed, pets, meat
Guinea pig (Cavia porcellus) USES: pets, meat, show, racing, research
Domestic rabbit (Oryctolagus cuniculus) USES: meat, pelt, fibre, pets, show, racing, research
Cat (Felis silvestris catus) USES: pets, pest control, show, pelt, research
Dog (Canis lupus familiaris) USES: Pets, hunting, herding, guarding, pest control, transportation, draft, working, show, racing, sport, rescuing, guiding, servicing, meat, research
Domestic guineafowl (Numida meleagris) USES: meat, eggs, pest control, show, alarming, pets
Mandarin duck (Aix galericulata) USES: meat, ornamental
Stingless bee (Melipona beecheii) USES: honey, pollination
Indian peafowl (Pavo cristatus) USES: show, feathers, meat, ornamental, pets
Egyptian mongoose (Herpestes ichneumon) USES: pest control, pets
Addax (Addax nasomaculatus) USES: meat, horns, leather, skin
Muscovy duck (Cairina moschata) USES: meat, feathers, eggs, show, pets
Red deer (Cervus elaphus) USES: meat, velvet, hides, leather, antlers
Western honey bee (Apis mellifera), including subspecies Italian bee (A. mellifera ligustica), European dark bee (A. mellifera mellifera), and Carniolan honey bee (A. mellifera carnica) USES: honey, wax, pollination
Fallow deer (Dama dama) USES: meat, hides, antlers
Semi-domesticated reindeer (Rangifer tarandus) USES: meat, milk, transportation, working, draft, mount, hides, racing, leather, antlers
Asiatic honey bee (Apis cerana), including subspecies Indian honey bee (Apis cerana indica) USES: honey, pollination
Asian elephant (Elephas maximus), including subspecies Indian elephant (Elephas maximus indicus) USES: working, transportation, hunting, show, racing, fighting
Scimitar oryx (Oryx dammah) USES: meat, hides, horns
For this list, I'm going to look less at individual plants, and more at centers of origin for plant domestication. Roughly speaking, there were eight "centers of origin" on Earth for independent rise of agriculture (two with subcenters), so the more realistic option would be to keep these various species of plants grouped together. So here are the centers, including where they were developed.
Center 1 (Mexico/Central America)
Grains and Legumes: maize, common bean, lima bean, tepary bean, jack bean, grain amaranth
Melon Plants: malabar gourd, winter pumpkin, chayote
Fiber Plants: upland cotton, bourbon cotton, henequen (sisal)
Miscellaneous: sweetpotato, arrowroot, pepper, papaya, guava, cashew, wild black cherry, chochenial, cherry tomato, cacao.
Center 2 (South America, northwestern region)
Root Tubers: potato, Other endemic cultivated potato species. Fourteen or more species with chromosome numbers varying from 24 to 60, edible nasturtium
Grains and Legumes: starchy maize, lima bean, common bean
Root Tubers: edible canna, potato
Vegetable Crops: pepino, tomato, ground cherry, pumpkin, pepper
Fiber Plants: cotton
Fruit and Miscellaneous: cocoa, passion flower, guava, heilborn, quinine tree, tobacco, cherimoya
2A (Chilean region)
Common potato (48 chromosomes), Chilean strawberry
2B (Brazilian region)
manioc, peanut, rubber tree, pineapple, Brazil nut, cashew, Erva-mate, purple granadilla.
Center 3 (Mediterranean Coasts)
Cereals and Legumes: durum wheat, emmer, Polish wheat, spelt, oats, sand oats, canarygrass, grass pea, pea, lupine
Forage Plants: clover, white clover, crimson clover, serradella
Oil and Fiber Plants: flax, rape, black mustard, olive
Vegetables: garden beet, cabbage, turnip, lettuce, asparagus, celery, chicory, parsnip, rhubarb,
Ethereal Oil and Spice Plants: caraway, anise, thyme, peppermint, sage, hop.
Center 4 (Middle East)
Grains and Legumes: einkorn wheat, durum wheat, poulard wheat, common wheat, oriental wheat, Persian wheat, two-row barley, rye, Mediterranean oats, common oats, lentil, lupine
Forage Plants: alfalfa, Persian clover, fenugreek, vetch, hairy vetch
Fruits: fig, pomegranate, apple, pear, quince, cherry, hawthorn.
Center 5 (Ethiopia)
Grains and Legumes: Abyssinian hard wheat, poulard wheat, emmer, Polish wheat, barley, grain sorghum, pearl millet, African millet, cowpea, flax, teff
Miscellaneous: sesame, castor bean, garden cress, coffee, okra, myrrh, indigo.
Center 6 (Central Asia)
Grains and Legumes: common wheat, club wheat, shot wheat, peas, lentil, horse bean, chickpea, mung bean, mustard, flax, sesame
Fiber Plants: hemp, cotton
Vegetables: onion, garlic, spinach, carrot
Fruits: pistacio, pear, almond, grape, apple.
Center 7 (India)
Cereals and Legumes: rice, chickpea, pigeon pea, urd bean, mung bean, rice bean, cowpea,
Vegetables and Tubers: eggplant, cucumber, radish, taro, yam
Fruits: mango, orange, tangerine, citron, tamarind
Sugar, Oil, and Fiber Plants: sugar cane, coconut palm, sesame, safflower, tree cotton, oriental cotton, jute, crotalaria, kenaf
Spices, Stimulants, Dyes, and Miscellaneous: hemp, black pepper, gum arabic, sandalwood, indigo, cinnamon tree, croton, bamboo.
7A (Southeast Asia)
Cereals and Legumes: Job's tears, velvet bean
Fruits: pummelo, banana, breadfruit, mangosteen
Oil, Sugar, Spice, and Fiber Plants: candlenut, coconut palm, sugarcane, clove, nutmeg, black pepper, manila hemp.
Center 8 (China)
Cereals and Legumes: e.g. broomcorn millet, Italian millet, Japanese barnyard millet, Koaliang, buckwheat, hull-less barley, soybean, Adzuki bean, velvet bean
Roots, Tubers, and Vegetables: e.g. Chinese yam, radish, Chinese cabbage, onion, cucumber
Fruits and Nuts: e.g. pear, Chinese apple, peach, apricot, cherry, walnut, litchi
Sugar, Drug, and Fiber Plants: e.g.sugar cane, opium poppy, ginseng camphor, hemp.
Hopefully, you'll find these lists helpful to build some interesting things. Good luck.