just dont get leylandii,they grow so quick but are a nightmare
cost me £800.00 to have mine removed there are lovely tree,shrubs and large bushes which will give you a better result in my opinion

Choose something that responds well to hard pruning unlike leylandii. If you want quick growers these will unfortunately be eventually very large (hence the quick growth, they go on and on...). Good hedging plants could be yew, beech, hornbeam. But for more choice shrubs etc. how about a trellis/hurdle-type of thing to give you privacy whilst they grow to the right size?

Trellis costs very little, especially if you construct it yourself from timber strips. You can then easily grow all sorts of climbers up the trellis, which will give you more choices than trees and shrubs and be quicker to establish.
There are so many clematis and other flowering climbers to choose from.
If you need something all year round, then choose ever green varieties.
If you don't mind waiting for something to establish, a beech hedge is a beautiful and permanent screen which retains its dead leaves through the winter.
Really quick growing trees include Ailanthus (tree of Heaven) which produces lots of new green shoots each year if cut right down. But watch for the suckering with this tree.
As others have said, all of these options are better than the dreaded Leylandii, or for that matter, laurel.

Beech does well on light soils but if you have clay, choose hornbeam rather than beech - both behave in the same way

i would be inclined while you have a blank canvas to grow lots of different types of shrubs,small trees and climbers that dont need so much cutting to keep the boarder more natural and informal.as bertie sais trellis is quite cheep you can even make your own to measure.if you have any animals you dont want to get though next door i wood just put some chicken wire up.this will have something for climbers to start of on and be hidden bye all the growth eventualy.have you seen what allen titshmarsh has done with a quite tall clump forming bamboo .he keeps splitting it and joing it on the end.he now has a fairly strate living fence that doesnt need trimming and stays fairly reguler.if nothing else if you have time i would research ideas and plants before you start.