On the left (west) side I would eliminate the border and extend the
lawn right to the hedge. That hedge defines a sharper, cleaner edge
than the border does, so the border there actually detracts from
the design value of the hedge. That's a fine hedge, by the way;
do you know how to maintain it? It will need regular clipping, a
nuisance job to do across a flower bed.

I would break up the lawn into "rooms", with a meandering passage
from room to room. That would involve a bed or two extending from
the hedge into the lawn, and from the fence into the lawn. Given
the hedge and fence, you have "partial shade" conditions. Not too
bad. Consider that when a vertical barrier behind a garden bed
is in shade, flowers on tall slender stems really POP. Think tall
varieties of poppy, daffodil, iris, tulip, daylily, etc. I have
daffodils where for several hours each afternoon the flowers are
in full sun but the wall behind them is shaded. The flowers look
fabulous.

Beds running into the lawn will get more sun than up against the
hedge/fence.

Against the fence I would grow climbing roses or other plants that
can make use of the fence and cover it.

How do you use the space? How do you want to use it? Go sit at
various spots around the garden, perhaps with a drink in hand,
and simply contemplate the space around you. Ideas will come to
you. Do you want it formal, informal? Elegant? Cosy? Spare?
Full? Low maintenance? Water conserving? Perennial?

I see pale cool colors in the bed under the red fence. Change
the color of the fence, or change the colors in the bed. Lots of
garden design books illustrate color combinations that work well.
I think those flowers would look better against the hedge.

Do I see a step in the grass back there? Someone is going to get
hurt.

I would move the rotary clothesline behind the cedar(?) so I don't
see it. Can you rotate the shed 90 degrees, so the door opens to
cement pad back there? Does it have windows also on the side now
against the hedge? I would rotate it and move it forward (south)
a few feet so it is up against the edge of the pad. I would have
a flower bed (more roses, very tall hollyhocks?) in front of it.
The roof of the shed will give that bed a little extra water than
other parts of the yard.

I'd remove the bush that is growing up in the cedar; the tree is
a very elegant plant, by itself. If you have any desire for a
little topiary in your garden, or some Zen rocks, that's where it
should go.

Una