I never "lug water" 'cos it's heavy and I'm old!
For the Cooler I connect it up to a garden hose and run the excess down the back of the garden.
For stuff that isn't going to be boiled (i.e. Strike and Sparge Water) I use a reinforced "food grade" clear plastic hose. (As a tip, sanitise this hose with StarSan before storage. I failed one time in the Spring when we went over to France for three months. When we got back I could see the black mould that had grown on the inside!)
I brew outside so I also bought a small low level trolley from Lidl that I use to transport heavy stuff to and from the garage. i.e. the filled Boiler from the garage and the filled FV to the garage. The distance is only about 15 metres but a Boiler with 30 litres of wort in it weighs about 32kgs and it's awkward to hump it without spillage.
With regard to plumbing in your shed, I sincerely hope that you installed drainage even if you didn't put in a water supply. Apart from the brewing, bottles need rinsing and washing, gear needs cleaning and rinsing etc. At the moment, all it would cost is a few bends and a length of plastic drainage pipe to the outside. A connection to your drains or a soak-away can be installed later; but doing it when everything is in place is much harder!
On the subject of water, after doing the maths (and having a row with the thieving swine that provide water to my house) I changed over from a metered supply to an annual payment rather than paying "Standing Charge plus Per Metric Ton of Water". The water company fought tooth and nail to keep me on a metered contract so I presume that they were making more money that way. For me, even if it isn't a cheaper option, it's much better because I don't have to worry about how much I am using as it's already paid for!
Good luck with the build.