I know a lot of these have already been said, but I'm going to list some anyway, because it's a fun question.
All we first had when I was a kid, I think, was this:
- some spades (you'll probably want more than one)
- a rake or two
- a hoe or two
- a couple hoses (one for the front and one for the back)
- a trowel or two
- a wheel barrow or two
- pruning shears
So, it's possible to garden with only that (or less). Of all those tools, I would have to say the hoses and spades were the most essential. You can do without the rest.
However, you might want to consider some more things:
- a weeder; this makes pulling up weeds a lot easier, especially if your soil isn't loose
- garden scissors
- One of those things you can fill with fluid for spraying it on plants (you might instantly think I'm implying it should be used for pesticides, fungicides, herbicides and such, which it could be used for, but you could also put water with other stuff, too; maybe even a neem oil solution).
- a fruit-picker (sometimes fruit trees are tall)
- special spray nozzles for your hose
Also, you might want to consider other gardening implements and products that might not be considered tools, but I'm not going to call any of these essential:
- A compost bin
- Containers for container gardening
- Black plastic
- Those things you fill with water and put around tomatoes to protect them from the frost (so you can plant them earlier)
- Tomato cages and other similar items
- Basalt rockdust (for some helpful extra minerals, if your soil isn't too alkaline already)
- Diatomaceous earth
- Stuff for vines to grow on (like a wire fence for grapes)
- Poles for beans
- a fruit-pitter
- something to test the acidity and temperature of your soil
- a water bath canner
- canning jars (lots of them)
- a storage room or two
- a ladder or two
- a pressure cooker (for canning things that aren't acidic enough)
- A rototiller
- A wood chipper/shredder (especially if you have lot of fruit trees)
- gloves; it's nice to avoid blisters and thorns, sometimes
- clothing fit for gardening
- LED grow lights (for indoor gardening)
- a pickup (great for transporting garden material, like if you need to get some sand or dirt and haul it to your house, or if you need to take tree branches away or something, or haul some giant pumpkins to your friends' house); also, they may be helpful in removing stumps if you have the right stuff; they may also be helpful for driving around your land if you have enough.
- Some large garbage cans (not just for garbage, but they're great for all kinds of garden stuff); I'm actually planning to use some for container gardening where the plants require deep soil, such as with melons (haven't done it, yet). I hear they're great for potatoes, though.