I’m actually doing something similar, but much more simple. Just wiring a light and receptical in a little garden shed we just bought. All I’m going to do is extend an outdoor plugs circuit that’s about 20’ away. No panels. I’m not worried about voltage drop and the fact that the circuit feeds more than one structure. I’m the inspector so no permit will be taken for this little job….lol.

My house panel has a 200 amp main breaker.
Here an outbuilding can have a main feed from there at 50% of the main so 100amps for me.
So I have a 100 amp breaker in the main house panel feeding a 100 amp main breaker of another panel in the barn.
(My barn is 100 feet from the house and I just ran the normal size wire for 100amps)
((Note the barn doesn't have everything on at once. I don't have the camper plugged in, air compressor running, be welding, have the fans running, and also be using the electric impact all at the same time.))
(((Meaning if you do then run the next size bigger feed wire and save yourself the headache)))
My panel in the barn is another big breaker box like in the house. This is where you want to spend an extra $100. Get the bigger main panel.
Then you have room from all the extra breakers you may need later. Even if you only put one breaker in it now.

If its a new build run conduit from the house to the barn and spend an extra $50 and throw in a few spare runs of pvc conduit to run things in later when you think of them.
phone line, data line, coax, flex gas line, air hose to the house, etc etc etc.
If you are pouring the floor use a drum or something and leave access to the dirt from the inside where you may want to pop up through sometime in the future.

Reading some of the comments for sure run ground (green insulated) white red and black from the house to the barn. You get alum cable from lowes for this run. Its a black cover with those four wires in it.
(this line from the house to the barn needs to be rated at the 100amps or whatever the main breaker in the house feeding it is, the number on it. If its 50A its 50A)
It gets some kind of dope on it so the alum conductor doesn't corrode in the panel. i.e. alum touching copper.
something I learned after... If you run ground (bare) white red black to all your lights and outlets in the barn you have that extra red conductor if you ever need it later.
i.e. two way switches or say giving each outlet in a two outlet box its own breaker... adding a fan next to a light.... a row of outlets for crocpot carry ins.

{15a breakers use white insulated 14awg nomex and 15 amp switches outlets light fixtures and etc.
20a breakers use yellow insulated 12awg nomex and 20 amp switches outlets light fixtures and etc.
100a breakers use 2awg aluminum, the ground can be smaller say 2-2-2-4. Use dope.}

I def drove a ground rod out in the barn.
The bonding screw in the panel on the barn is a local issue it depends what your inspector wants.
Ag buildings here don't get an inspection and I figured it best to not have anything floating but I'd have to look if I connected it or not.
My barn is metal pole barn and I grounded the metal of the barn to the ground rod too. I didn't want it to float either.
(i.e. have a wire short to the barn making the barn metal hot and touching it on the outside and the ground coming through your feet to the metal you are touching)
((have it connected to the ground rod so it trips the breaker not floating there hot waiting to be touched))
Also run all metal gang, switch, and junction boxes so that bare ground wire can be connected in each and every box you hang.
The pvc ones look handy but pvc isn't a conductor and there is no way to connect the bare wire to it.
(When there is a short you want the breaker to trip not floating in a plastic box especially in a metal barn.)
((this becomes important when you start drilling holes in metal siding to hang outside lights))
(((or running your wire along the wall and an accidental short hitting the barn metal)))

Also realize that if you gen outlet is inside your separate outbuilding that when you go to start and run your gen it can be inside then not be fooling with it in the snow and ice.
Then you realize you can install emergency lighting that has a battery and kicks on when you lose power so when you go out to hook things up your gen spot and panel are lit up.
i.e. instead of feeding 100amps from the house to the barn kick on the gen and run 100amps from the barn to the house (after you disconnect the house main from the main road feed)

This isn't advice for what you should do. This is just what I did. Hope it helps. Good luck.

We went thru something similar last summer. Old pole barn that had been built and wired about 70 to 80 years ago. Wiring then very creative, but had worked for all those years. So last June big wind storm blows thru and and a fallen tree limb takes out the overhead lines........and it being a live wire with no shutoff or breaker from meter to the barn, caught the ground on fire in the process. Not good. To shut it off, power company had to be called and to shut it off they pulled the meter. That meant "grandfather" status died.

Overhead service was a single pair of # 6 coated hot and uncoated neutral that ran from meter pole to terminal at top of barn gable. There, two #10 hot wires hooked on to the one hot overhead. Those then ran to a two outlet fuse box........each 15 amp 110v circuits. One hooked to some overhead lights......the other to about 4 outlets. In many ways, the hinky little fuse box prevented anything really wild from going on. Still......there was some other curious stuff. Switch for the overhead lights by a pedestrian door. To wire the switch, they ran a single romex wire from a junction box to switch box about 40 feet away. The black or hot wire in this single wire was the supply to the switch.........the white or what would otherwise be the neutral wire was the return (it was also hot when the switch was on). I doubt any of that to code, but it worked (I should say works, as it is still wired that way)

But to get service to the barn up to modern standards to satisfy power company and local building inspector, we had to bury 4 wires in the ground.......two hots, neutral and ground. The new panel in the barn is now a sub panel. No ground in the barn. Ground has to run all the way back to the ground rod at the meter pole. Outlets now all on a GFI breaker. No plans to add more circuits, but now could if someone wanted to.

Modern service done right is probably the way to go.....but is not the cheapest way to do it. But getting someone fried or burning the barn down is not cheap either.

Edited to include........we also installed a breaker box at the source on the meter pole..........which allows the whole circuit line to the barn to be shut off if need be. Service panel box in the barn does not have a main breaker. Only circuit breakers.

Yup, that's the way to do it. So again, I suggest 10/3 with ground for 20 amp 240v service, allowing two 20 amp 120v circuits (each can pull the full 20amp).

"10/3 with ground" I suggested before is 4 wires. Thats two hots, nuetral, and ground. The neutral is equal in size to the hots (not double the ampacity) because if one leg is fully loaded (20 amp) the nuetral carries 20 amp. But if both legs are fully loaded, the nuetral carries nothing. In practice, the neuetral carries the difference betweent he two legs.

When running power to an out building, if you're absoulutely sure all you'll ever want is a single 15-20 amp circuit, then run that. But if want anything more than a single 20amp circuit, run it in 240. Adding a single conductor doubles the usable power.

Yes. 4 wires are required. Do not share the neutral and the ground. I see people doing this all the time and it's wrong. Also, search "southwire voltage drop calculator" on the web. It will allow you to put in your specific parameters ( distance, current, etc.) and give you the correct wire size.

To wire to NEC standards... you are suppose to have a disconnect on that shed and a ground rod. I would suggest installing 3/4" pvc buried 18 inches deep from your 30 amp house feed junction box to the shed disconnect.
Install 4 to 6 circuit panel back of disconnect and feed your circuits. Receptacles are suppose to be GFCI protected.
I would have 2 or 3 dedicated 20 amp GFCI receptacles on #12.
30 amp 240v 4 wire system should be plenty of power for your needs...you won't have a significant VOLTAGE drop..no NEED to run #8.
There's only one way to do it..the right way. We fix hacked up electrical jobs often and it costs the customer money.
There's a reason electricians are licensed professionals.

Best answer posted yet! Thank You rosscoak! (and yes I'm a retired Master Electrician).

I just ran power to a shed. Passed inspection with flying colors. It required 2 grounding rods 6’ apart.

The exception to 2 rods is if you have 25 ohms or less ...only one rod is required. A ground resistance ohmeter is required to show that number

Which is probably why guys just pound 2…….or here we can use one plate instead of 2 rods. In the gumbo soil 25 is achievable, but I remember us trainees pounding 7 rods on our first crew. The instructor would roll up in his truck, toss out the tester and say “tell me a good number”. Go to test and it’s reading 44 ohms…….”21 ohms Chris!”
There’s a town close by called Sandilands, built on sand of course. Voltage fluctuations for years. We hired a contractor to come in and pound new rods to try to get each pole ground down to 50 ohms. The average is around 70ish ohms, some rods were stacked over 100’ long. Would’ve been easier to just run a 2/0 ground along the line under the neutral, but then the engineers get involved and the 5 miles of Y line would have to be rebuilt to accommodate this…..lol