I never use cone wrenches any more. And there's lots of different bottom brackets and crank tools. You want to get the correct ones for your bike.
Most of my tools aren't in these sets!

For my bike: (We'll see if other commenters have different ideas.)
I enjoy working on my bike. So I buy tools as needed.

A repair stand. They are expensive, but last forever. Every repair or adjustment is easier with one.

Chain breaker tool and quick links to rejoin the chain. New 10 or 11 speed chains are fussy about installing the new pin correctly. The quick links avoid all that. I've used Connex 10 speed links, which are strong and very easy to connect and disconnect.
A good metal ruler for measuring chain wear. I have one of the chain checkers, but the ruler is more accurate.

Cable cutter (including a crimp end tool--most of these do.) and a bag of cable crimps.
Dremel to grind off the ends of the cable housing. You might be able to use a file, but that's really slow, and a Dremel has lots of other uses.

4-5-6 mm Y wrench
A set of hex wrenches--the loose L ones work better- good leverage and can fit into tight places, but a set that folds out of a handle is okay.
I also have a set of 3, 4, 5 mm hex screwdriver handled, so I can start and spin on the bolts easier.
Yes--I have 4 different types of hex wrenches, including the screwgun hex set! But I started with a set of L wrenches.
Don't get the "dollar store" cheap no-name hex wrenches. A good set will be accurate and won't round the edges off.

A bottom bracket tool and any tool required to remove your cranks. It depends on your cranks and bottom bracket styles. Sometimes the crank area gets creaking or clicking noises, and it's good to disassemble, clean and grease it.
Or you might remove the crankset so you can reach in and turn the bearings by hand to see if they are worn out. They may be press fit to replace them, and that's a good job for a bike store.

Pedal wrench. My pedals use an 8 mm long handled hex wrench, there aren't any of the usual 15mm flats on my pedals.

Black handle Park Tool spoke wrench. This is the correct size for my spoke nipples.
I have a Park Tool tension meter and a truing stand, since I like working on wheels and have built a couple of wheels. Optional otherwise.

Torque wrench. An inexpensive 5 nm preset one can work okay, but I like the Park Tool beam wrenches. I have the smaller one, from 0 to 7 nm. Mine needed separate adapters to fit screwgun ends. I have a set of hex ends to use with the torque wrench.
Carbon assembly paste in a tube. This works great on seatposts and on handlebar clamps, even if they are aluminum. Much less torque is needed to hold it in place.

Blue Loktite thread locker. Mostly for bottle cage bolts and tail light bolts. I use grease on stem and seatpost bolts.

13mm sears open end wrench that fits the flats on my brakes, to hold them while the 5mm wrench tightens the bolt.

Cassette tool. Mine has a 1 inch hex outside, so I got a cheap 1 inch deep socket to fit it, and I can use a socket wrench. I already had an automotive sized torque wrench, so that's good when tightening the cassette, but it's probably not necessary.
Chain whip to use when removing the cassette.

Misc:
small flat blade screwdrivers.
small Phillips screwdrivers.
Torx bolts are becoming more common! I use a screwdriver handled T20 for my Campagnolo brakes. And I think a T25 is quite common, but check your own bike.
Diagonal cutters. Mostly for clipping zip ties. Terrible for cutting cables.
Pliers.
Flashlight.

3M electrical tape. Way better than the no-name tape.

Silver Sharpie pen for marking dark surfaces. I use this a lot! (rubbing alcohol takes it off)
bike grease in a tube.
Chain lube.
car wax for the paint.
90% rubbing alcohol from the drugstore for wiping off the chain, etc.
Odorless mineral spirits from the paint store for cleaning cassettes and getting grease off.

Keep a spare tube on hand. Keep an old worn tire on hand to use temporarily in case a tire is cut too bad to ride.
Tire levers
Floor pump

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To start:
L hex wrenches.
grease.
carbon assembly paste.
metal ruler.
chain tool and quick links.
spare tube.
tire levers
floor pump of course.
misc screwdrivers.
One of the comments above mentioned headset spacers. I've been trying different stem heights on my steerer tube, so a couple of spare 5mm spacers is good to have, to try different setups.

Add a torque wrench if possible. It's getting more critical. Most new mechanics aren't good at estimating the correct torques. EDIT--I've seen two different threads lately where the carbon steerer tube was cracked from too much torque on the stem bolts. At least get an inexpensive 5nm torque key.

If you like working on your bikes, get the repair stand. It'll really help.

Then these can be added later:

new shifter cables after two years. The back one tends to fray where it bends at the shifter. And new housing will help it shift smoother. So get the cutter, the cable set, and the crimp ends. A dremel really makes this easy--the cable housing is crushed or bent where it's cut by the cutters. Grinding it off flat with a Dremel is easy.
I'm picky about getting the cable housing just the right length. Bikes come with extra long housing to accomodate different bike setups. When I cut my own, I can go back and trim another bit off until it fits perfectly.

Creaking or cranks that don't spin easily: either take it to the bike shop, or get the tools to disassemble. My old bike had recurring creaking that was easily fixed by pulling the crank, wiping it off, and regreasing.

Wheels a little out of true, with brake pulsing: A spoke tool.

New cassette, either worn out (after maybe 3 to 5 years? ) or a different size wanted. Cassette tool and chain whip.