If that were my space, I'd start by raking out as much dead grass as possible before thinking about mowing. If you're not sure if there are spring bulbs in the lawn, at least cleaning the dead grass will give you a better look at what's below it. If you don't see any signs of bulbs coming through yet, then I'd mow it and leave it for a while to see if anything pops up in late spring. Anything you can positively identify as a weed can be dealt with now, while the soil is easily workable (if it's not too wet). You never know, that lawn may be full of wildflowers. You might ask your neighbors if they know anything about the garden, too. Maybe they've seen it years past and can give you a little guidance as to what might be growing in a few months. The fences, path, and that gate are just beautiful, you have a great little space there!
New England, USA
Metacomet soil with hints of Woodbridge and Pillsbury
If there are borders round the lawn, cutting new edges will make it look instantly tidier even if the grass itself isn't looking great yet. You can do it with a flat-bladed spade if you don't have a half-moon edging iron. For really straight lines you can put something like a plank or scaffold board down to use as a guide.
Doncaster, South Yorkshire. Soil type: sandy, well-drained
Welcome to the forum @cloeandrews73357. Will watch with interest as you progress as it looks like you have the bones of a very pretty garden there. And that looks as if it might have been a rock garden on the right of the picture.
How does the sun behave? At what time of day does the sun hit the old rock garden for example? And do you know what kind of soil you have - heavy and clay-like, or light and sandy? All this will help inform your planting choices when you get to that stage.
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How does the sun behave? At what time of day does the sun hit the old rock garden for example? And do you know what kind of soil you have - heavy and clay-like, or light and sandy? All this will help inform your planting choices when you get to that stage.