I love gardening but haven't been able to for ages cause of pregnancies etc so here's some tips from a frustrated gardener who would love to be sorting out a new plot like yours!
Always plant in odd no.s - 3's or 5's etc otherwise it looks too artifical.
Think about height, gardens are as much about up as across, the silver birches are a great idea. With trees the smaller you plant them the quicker they are to establish, birches are particularly quick, in a few yrs they'll look like they've always been there.
Too stretch budget buy mail order, you'll get far more for your money and avoid making on whim purchases at nursery.
Pick up a few copies of Gardeners World, always full of great ideas. Plus you never know, you might really get into it, gardening can become a passion.
Some grasses and allium bulbs around the birches would look impressive but is simple to do.
Don't neglect shady spots, there's a huge range of ferns, planted in groups they always look v rich. Try some Hellebores too.
When planting always remember a penny for the plant a pound for the soil - I think it goes, anyway as you plant enrich the soil with compost, some chicken pellets, or a sprinkling of seaweed feed, just to get the plant off to a good start, then water.
Otherwise gardening is really easy, just read label on plant and always put plant incorrect position, sun, shade etc, your garden will oblige and become beautiful.
If you use pots, get the biggest possible so that the plants aren't too dependant on you for watering and use moisture retaining gel. Pieris look amazing in spring time in barrels, you would need to use acid compost. Mostly though container gardening isn't a quick fix but labour intensive.
Grab a few gardening books in a secondhand store and you'll have it soon figured.