Ollie,
I would suggest that unless your grass is dying (because of a disease) then you should look to repair/ replace in the most economical way, this would generally be through allowing new growth to come through - seeding. But use a decent seed supplier. Quality (fitness for purpose) is key.
But you need to start now Ollie, if there are some good areas of grass, then start to remove the weeds now, spring is a good time for this. And following a hard winter such as the one we've had, scarification is vital. You need to remove the moss that has been allowed to grow, rake out the dead/ dry grass.
Only once you have put this hard work and effort in should you seed, it gives the best chance for the seed to develop.
However, before looking at a complete new lawn solution, you need to consider how you use your grassed areas. Are they subjected to heavy wear? Are they allowed sufficient sunlight, watering etc? What about the recovery time, do you have times where you can carry out maintenance and allow new growth to develop.
This will help you decide:
a) The type of grass to use (there are a great many types and blends, go for the best quality you can afford)
b) the old Turf v seed argument.
All of these decisions you must take with an intelliegent, informed review, perhaps invest in the use of an expert to give you advice. It could be crucial.
Then you need to nuture your choice, if its seed, invest in some protection, there are birds awaiting to pick the new seeds, you don't want to see your very good quality seed being picked off too soon.
If its turf, don't just chuck it down and hope that it takes and matches with what you already have. You need to constantly keep attending to new turf as well, I know its expensive, but perhaps even chuck some seed in between too.
If you have some spare land/ areas which aren't used frequently, perhaps seed these areas, you can slowly develop good healthy turf of your own and those areas where you see are struggling later this year or early next, then you have a ready grown option.
Perhaps ask some friends to donate some of their land to allow you to grow some extra turf. If its good stuff and you don't need it, then you could always sell it off or let them have it for another season or two.
Follow these tips and you'll reep the benefits of a healthy lawn to look out onto from your shed