Since each person has a different income, how much each person each person should save will vary greatly. Each person or household will have different financial obligations. There is no specific dollar amount that will be suitable for everyone's situation. You need to setup your household budget first, then make the decision on your savings goal.
There are some basic guidelines you should follow when making your decision on how much you should save each month.
The 10% Rule
The standard is 10% of your income as a starting point. Now granted, you may not be able to afford to save 10% of your income but you want to work toward it.
Don't Make it Painful
In the beginning you don't want to save until it hurts, it will cause you to get discouraged and quit saving all together. You want to lighten up a bit so that you have some breathing room in your household budget while still being tight enough that you have to watch your spending.
Once you begin saving your money, you need to set goals for that money. Is that money being set aside as an emergency fund or for a new car when the time comes. It makes it easier to save if you will set a purpose for that money. You now have a "vision" for that money and it's plain.
Increase in Time
You should be able to increase how much you save over time since you are living frugal and working your debt elimination plan. You should start seeing your debts come under control. If you can increase your savings to 20% per month by all means, do it. You also want to watch and not neglect your retirement fund. If your employer offers a 401K match, take them up on it and contribute to the plan to take advantage of their match. The company match is free money.
Now that you are living frugal and being diligent about saving each month, you are starting to see the light at the end of the tunnel. You can do this, all it takes is perseverance and discipline and you have both.