To get started, you first need to get in the mindset of saving. If you have any debt, it's best to prioritise putting aside money to pay it off as soon as you can. The interest rates on your debts will typically be higher than the interest you'd get by saving, so you'll be saving money that way.
Once that's done, you can focus on saving. A good target to aim for is putting aside 10% of your income.
10% of your income may seem like a lot of money, but if you can find a way to cut back and live on 90% of what you earn, you'll have more available for your future needs.
If you're new to saving, it helps to focus on the positive aspects of saving money rather than the pain caused by not being able to buy everything you want right away.
Instead of thinking about what you can't have or what you need to cut out of your life, think about the things your savings might enable you to buy — a better car, a holiday away, or tickets to a big game.