Now of course they want privacy and yes you guessed it a garden that feels established and mature. I wish I had a pound for every time.
I always get the same crestfallen expression when I explain that the only way to replace those overgrown shrubs is time (often in years) and then shock when they realise just how many plants (and £s) it will take to refill their garden.
Its always worth taking time to get to know a garden. Maybe those trees and shrubs are there for a reason, providing privacy or shade or maybe fabulous flowers all summer. Try and get advice from someone who knows about gardens and gardening. They can hopefully guide you as to what is worth keeping and what needs to go, but always be aware that even the best gardeners come with their own baggage and likes and dislikes. If they are saying rip it all out, ask them what do you put in its place and how much will it cost. If they cant give you an answer then their advice might be best ignored.
A weed filled lawn can often be rescued by simply mowing regularly and some simple TLC. The weeds will give up and the grass will recover. Maybe not to bowling green standard but good enough for most of us.
Take stock of the garden and make a plan. Don't strip an area bare until you know what you are going to do with it. As my young couple found out, brambles and weeds love an empty bed. You might not like the shrubs you have inherited but they are better living companions whilst you introduce your own choices.
It might also be worth keeping some of your existing planting whilst your new plants develop and grow. You can then thin and remove as needed over time. Not all plants need to be the star of the show to perform a role and keeping some of the existing plants will give your garden an established, mature feel. It takes time for trees to mature and the one you inherited may not be the one you would choose but take a moment. Do you intend to live their for the 10 years it may take for your new young tree to match the image in your mind? If yes then go for it. If not do not underestimate what that boring tree might be doing for you and what you might miss when it's gone.