Honestly? Well first of all don't have any major fixed expenditures in your budget. If you already commit 25% of our income to housing, another 25% to cars, another 25% to taxes, you'll only have 25% left. How can you save a lot when out of that 25% is your living expenses? And some of those expenses will also be fixed, such as life, health, care insurance, utilities, and groceries.
The best way to accelerate savings is to lower your fixed expenses. That way you have the flexibility to increase and decrease your saving ability as needed.
Another reason to do it early, in your 20s, is that you'll grow into your income slower. You won't get use to spending so much. That way you'll have the flexibility to go back to school, change careers, stay at home with kids, or not work if you so choose.
Savings gives people flexibility of choice. So accelerate your savings by living below your means, and keeping your fixed expenses low.