Totally standard thing to do.
My exterior workshop has power - MANY sockets and lights.
No 2 son's garage has several sockets and lights.
No 3 son built an outside utility room with sockets and lights.
Cost depends on many things unfortunately:
Where the Consumer Unit is located.
What type of Consumer Unit and if it has any spare capacity.
Ease of access to Consumer Unit and from there to outside .
Distance to outbuilding.
What you want in the outbuilding - how many lights, sockets, external lights and sockets etc.
Can you get some-one to dig the trench and clear the shed and area around the Consumer Unit? Unskilled 'free' labour to avoid paying a tradesman's rate.
So from a couple of hundred to a thousand.
Unlikely to be more than one thousand unless:
You have a very old house with a very old Consumer Unit - for example with rewirable fuses. An electrician will want to change that for a modern one with all the modern safety features. That would add a couple of hundred to the bill.
Concrete or tarmac between house and outbuilding. Digging that up and relaying could be hundreds
A VERY long way to the outbuilding.
You want many lights and sockets or 'special' features.
Write a list of what you want, take photos of your Consumer Unit, measure rough distance and note what is 'in the way'.
Then Google or Facebook local electricians, describe the job, ask if they are interested and get 3 quotes.