New gardener but with very mature garden

[miserablecat]

My parents loved gardening, but I've never been especially interested.
When we moved into our house over 10 years ago it had an established and mature garden. DH put in a pond and has made some vegetable beds at the end. I've finally thought I might like to grow some flowers but it feels like there's no space and the lawn is bordered by shrubs bushes etc so it feels very claustrophobic, not helped by the fact we've not really kept on top of cutting back etc. I try to prune them (nowhere near frequently enough) but they are so dense and it feels like there is literally no bare soil to plant anything.
This is a stupid question but how do I create space? They just seem to have so many branches etc. Would I need to dig them out/get professionals in? What happens to the roots? Pics to show how wild it is!!

You need to have a very serious cut back. Start with one shrub which is very overgrown. Take out up to one third of the branches right down to ground level, to make it less dense. Then cut the remaining branches back to a framework slightly smaller than what you would want it to be.

Then gradually do the same with each of the shrubs and see how it looks. You can then get an idea of which shrubs, if any, you would like to remove to make room for perennials and other smaller plants.

Start by digging out all the remains of the borders. You have what looks like ground elder in the first picture, you’ll want to dig out all the roots of that. Start a compost heap - you don’t want all the greenery to go to waste. Most of the plants you pull out can go straight on the heap, but any thick fleshy roots leave out in the sun for a few days so they’re properly dead before adding to the heap. Next year, you’ll be able to take the half-rotted stuff off the top of the heap, and spread the compost underneath on the borders. Put the half rotted stuff back on the heap.

you can do light pruning in the summer. If you do it just after flowering, you'll still get flowers next year, if you’ve left it too late, you won’t harm the plant but you may miss a year’s flowering.

renovation pruning - leave till winter. You can cut up to a third of the old stems out completely, at just above ground level. Then repeat the following two years, ie follow @CatherinedeBourgh . Google renovation pruning of apples - the RHS has a good set of instructions.

your second picture has what looks like winter flowering jasmine. It’s a bit entangled with the shrub but tolerates hard pruning.