The first step to take is work out how much you will need. The minimum for a mortgage deposit in the UK is usually 5%, although be aware that the coronavirus pandemic has caused financial uncertainty, and some lenders are asking for at least 15%. Remember that the larger the deposit you put down, the lower your mortgage interest rate will be.
Take a look at property and estate agent websites for the area you want to live in. Decide if you have any essential criteria for a property, such as three bedrooms, or to be within a certain distance of local schools. Note down the asking prices for all the suitable properties, and then use a borrowing calculator to work out roughly how much you need to borrow.
To help save money, put together a savings plan. Do some budget maintenance, and cancel unused subscriptions, switch to cheaper broadband and energy suppliers, and make sure you are paying the correct amount of council tax, especially if you live alone and are eligible for the 25% single person discount.
If you are currently renting, you could consider moving to a flatshare or bedsit, or moving to a cheaper area. Some people will of course have the option of moving back in with their parents for a while. You could also look into increasing your income, through freelancing, selling any unwanted items, or applying for a promotion at work.
It is helpful to set up a direct debit which will move money into your savings account on the date you are paid. Shop around for a high-interest rate savings account, or set up a tax-free ISA, such as a Help to Buy ISA or a Lifetime ISA.
Remember that a deposit is not the only upfront cost of purchasing a home. You will also need money for surveys and conveyancing, building insurance, land registry fees, removal fees, and stamp duty.