I have a monthly budget to estimate my expenses. What are good ways to make sure I stick to this budget? Would a savings account help?

First of all, congratulations on being mindful of your monthly expenses/savings. If it doens't already, your budget should include your income as well as your expenses. Once you have totalled each of those against each other, see what you're left with. Assuming your expenses are below your income, can then decide how much of the excess you want to save, and how much you'll allow yourself to spend on miscellaneous items. 

Let's say after you pay all of your bills, you're left with $600. You could choose to put $200 in a savings account, and keep the other $400 for yourself. Consider taking out $100 cash at the beginning of each week. This will help you to keep track of how much you have left, and prevent you from spending beyond your budget.

For expenses that fluctuate, such as food, or clothing, you can set a limit at the beginning of the month of how much you're able to spend. As you make purchases in those categories, write them down, so you can see your progress toward your allotted amount. At the end of the month, if you've stayed under the allotted amount, you can put the rest into savings. 

It is never a bad idea to have a savings account. If you are making regular contributions, you will build up a nest egg to use in the event of an emergency, or to make a large purchase using money out-of-pocket, rather than financing.