I'm a new homeowner and as expected, the developer stripped off all
the good topsoil before starting this subdivision. My yard has a
couple inches of decent topsoil, but below that is nothing but clay.

This is more and more common. When buying a newly built home it's usually
recommended to follow-up on this 'un before it becomes an issue. (Live and
learn, right?)

I've managed to get a lawn seeded and it came in pretty well, but it
is patchy in some spots. I'm assuming this is areas where there is
less good topsoil.

This may or may not be the case -- just dig down in the bare spots a little
bit and you'll know for certain. You may need to do a little soil
replacement in these patches by just digging out the old and replacing it
with new.

Is there anything I can do to improve the soil and get the grass to
root deeper?

If you want an immediate solution then you'll have to dig up the property
and replace the soil or buy a few tons of topsoil to put over top of the
existing, amend it, and re-seed. Not very practical of course. The best
solution though...

Organic material, organic material, organic material! Compost is your
best bet here -- you can buy bags of it and simply spread it throughout your
lawn, use a mulch mower and leave clippings where they lay, set up a spot
for a compost pile and/or bins (leaves, table scraps, etc) and so forth.
This does take a while, but over time it will do the trick very nicely.

I'm also having a hard time trying to get any kind of tree or bush to
grow without putting in a HUGE hole full of decent soil.

Well, not much else you can do here -- soil replacement is the way to get
these in and growing healthy now plus the addition of organic material to
the plot in general so when the roots spread they'll find a good nutrient
base.

Aside from grass and trees, you can also use raised beds as a landscaping
technique. Limestone beds are currently very well favored among quite a few
folks, and even concrete and wood will work nicely if you find a design you
favor. The nice thing about raised beds is that YOU control their soil
content completely. If you have a vegetable garden, this is the only way to
go.

A simple raised bed design uses nothing more than three 2"x8"x8' boards and
some deck screws. You cut one board in half to make the ends then screw the
4 pieces together. Put it in a sunny spot and fill it with soil & compost.
Instant perfect vegetable gardening.

James