Looks as if you're new to the site too, so welcome to GoY.

I hope you've also booked to have the stumps ground out or removed from your trees - if not, best do so, or you won't be able to do much with the area. Assuming the stumps and roots are going, next thing is to tidy it up (it'll look a right mess afterwards) and then get some composted animal manure or similar, spread it over the area, then fork it over. You need enough of it to make a minimal 2 inch layer across the whole area, prior to digging in. Leave it to settle for a couple of weeks while you think about what you want to plant there instead.

If you're sure you still want a hedge, you've got several choices - you could have an informal hedge, which means you just choose 4 or 5 five shrub varieties (depending on the length and width of the border) and let them mingle freely together, occasionally trimming if it looks a bit untidy. Otherwise, a formal hedge, which you keep trimmed with a hedgetrimmer a couple of times a year. You also need to consider what height you'd like the hedge to be before making a final choice.

Once you've decided those things, then its time to ask for options as to which plant/s would suit. The amount of sun the area gets is also important, as is how exposed the situation is - there's a big difference between a south facing garden and an east or north facing one with nothing opposite your house. Where you are, that could mean very cold winds in winter, so that information would be useful too. If you're not far from the sea, that would be useful to know also.

Great advice from Bamboo.

I'd add one thing. If you are a nature fan and you want to encourage the birds into your garden you could always opt for a wildlife hedge. A google search will show plenty of information and options for this.

We have a mixed shrub border as a short hedge , its pretty and the birds likeit.....have you looked in Goypedia for hedges?
use the search box above

I'm of the opinion that a mixed hedge is alright at the back of a rural garden but can be untidy as subjects grow at different rates. For a front garden then (Taxus) Yew makes a super formal hedge; far better than Buxus as not affected by blight or Box aphid. Just to add, take the good advice regarding the removal of the conifer roots as it is false economy by leaving them in.

Yes, mature yew hedge is good, but its very dark unless you get the golden one, and it does grow slowly. We need to know how high Daisy wants it to be too . I'd hasten slowly and look at lots of different sorts - you don't want to rush into things and then wish you'd had something else. You have some thinking time as it will take a while to get the soil into good heart after those conifers.