A starting point would be to load up everything you bought and take it to returns at home depot (: Sorry, couldn't help it but that just reminds me of all the customers I've had with a bag full of blue boxes, switch plates, and random fittings saying "I've got all the material for you". Your job is somewhat difficult as far as the correct materials. The maximum length for the liquid tight flex is 6 feet which you are a bit over for your run. It would be very interesting to see much of any wire make it through ~70 feet of liquid tight. The burial depth is 12" if it is a GFCI protected circuit. 18" would be for conduit without GFCI protection, 24" would be for UF with no GFCI protection.

Not sure about the little green box but it sounds like something that is made to be cord and plug connected? Really do not need it. Just use a metal 4x4 box with a mudring for a GFCI or a mudring with GFCI/ switch.

You will need to buy some thwn wire (the individual black, white, and green) in whatever length, if you decide to run pipe. 3/4 or 1" pvc would be good. UF is really not supposed to be used in pipe except to protect it as it comes out of the ground. The individual wires will pull easy and be suitable for underground. Into the shed you will either have to pipe directly in with an LB fitting heading to your box. Or if it gets tricky, end the pipe and wires in a weather proof junction box outside and come into the shed with MC, indoor NM cable, or some liquid tight (6 ft !) At the house, just stub the pipe up to your outdoor outlet and put a weatherproof outdoor box over the existing outlet box (the kind with an open back). That is unless your outdoor outlet already has an exterior box (sticking out from the house).

I believe most cities would be happy to sell a permit for as little as changing a light switch. In my area it seems that permits are pulled for remodels (except for non-permited remodels like 50% of them are) or for anything involving the power company (such as a service upgrade), or of course new construction (whole house, large addition). I suppose everyone is breaking rules but I really don't know of anyone that goes to pull a permit when someone is putting in a new circuit for their freezer or adding 10 recessed lights to their ceiling.

"I've got all the material for you". bwahhh ha ha.. yeah tell me about it...

thanks! so HOW DO I KNOW THAT THE OUTSIDE
receptacles (and in turn the wires run to the shed) are in fact gfci protected??? breaker in the garage is labeled "gfi garage" and the outlet in the garage is a gfci outlet.
but the one on the outside of the house that is also controlled by this breaker looks like a normal outlet...

I think the green box would still work, its meant to be wired in not plugged in has a 1/2" or 3/4" knockout on the bottom... and instructions tell how to wire it up... after its wired in you have 2 gfci receptacles and a switch to turn it off and on (and heck I did pay 30 bucks for it lol)

the extra liquid tight will be good for drainage for my landscaping later I guess :)

I like the 3/4 pvc pipe and the individual thwn wire (the individual black, white, and green) run should be easy if the burial debth is 12 inches and it's ok to come up through the floor of the shed. straight then 90 degrees up the floor to the box, straight out the other end to the box on the house up 90 degrees... piece of cake...

thanks! I pushed test on the GFCI outlet in the garage and it turns power off to the outlet outside. So looks like I would only have to bury the cable 12". thats good cause 12" I can do myself, 18-24 I would have to hire someone... too much limestone here...

Rigid or intermediate metal conduit only requires 6 inches of depth.

It runs about $1 a foot, but only needing a trench 7-8 inches deep (the burial depth is to the top of the conduit in the trench) makes it often easier and faster to install (even with a ditch witch).

You might want to consider just putting a sub-panel in the shed.
That would allow you to have more than a single circuit.

ahhh so the burial depth is to the top of the conduit...
that rigid sounds even better... worth the $1 a foot if it saves me from digging 13-14 inches....

going to call the city and have them come out and map my utilities first so I don't hit anything white digging and then I'll go from there...

thanks to everyone that gave me info! really helped allot, now I can talk to an electrician with some idea of what I want to do...

Finished up a rental house repair a few months ago.
About 35 feet from the house to the light pole.

While digging for plants a previous tenant cut the UF cable that the previous owner buried about 6-8 inches deep.

It took about 2 hours to trench out, install rigid, pull wires, and have everything back up and running.

Only needing a 7-8 inch deep trench makes short work of things.

No ditch witch rental, two people, two hours, done.

The plugs of grass were put along the trench (if you can really call it that) on one side and extra dirt on a tarp on the other side.
Everything went right back into place.

If you look now you cannot even tell were the work was done.
no grass seed, no sod, nothing special except telling the tenant to water the area every evening.
Since I pay the water and trash anyway, they did a great job.

thanks, my grass is all dead so that part is easy, good tip on the tarp though, was thinking about that today... tarp is perfect!