If you are struggling to just make a security deposit, you will struggle with all expenses, there will always be additional expenses, so you need a cushion to fall back on in savings.
Utilities: gas, water, electricity, sewer bills. If the apartment doesn't include them in rent, you will have to put up deposit to get utilities in your name. Telephone, cable TV, internet. These can add up very quickly, ask your parents or friends what their bills are, to see if you can afford.
Besides your kitchen supplies listed in another post, covered most things.
Don't forget cooking oil, baking soda, recipe book, spices. Cooking is cheaper than eating out and healthier. Things like plastic wrap, aluminum foil, garbage bags optional, (grocery/Walmart bags work too if you have a dumpster close by).
(For me) SHELF LINER /drawer liner is a necessity, after I clean the cupboards and drawers BEFORE moving in or unpacking.
Vacuum cleaner.
Cleaning supplies, toliet bowel cleaner, toilet brush, disinfectant. You can save money by using rags instead of paper towels all the time. They will wash and can be reused. Rags for cleaning windows, mirrors, old soft worn out T shirts work great also for dusting, they don't have lint.
Shower curtain
Laundry money if you don't have own washer/dryer. Laundry supplies, detergent, softener, or dryer sheets if you use them.
Don't forget Insurance.........renters insurance (recently in Metro KC fires destroyed many apartments or had damage, and those interviewed on the news did not have insurance to have anything to replace anything they lost). Check around with different companies, could be $100 or so per year for about $10,000 in coverage.
With a child, you should have insurance, on health and life.
For transportation, gas money, oil changes, tire rotation, maintenance, insurance.
If you are going to live on your own for first time, and you shoiuld, there will be unexpected things that come up all the time. With a growing baby, think of child care, dr. visits, pre-school.
Learn how to save, re-use, share. Not everything has to be new.
Estate sales offer whole house-holds to choose from, as well as garage sales, great buys!
I have found food to furniture, tools to clothes at estate sales. Toys, desks, cleaning supplies, pictures, frames, linens, stationery, cards, many surprises.
Best of luck on being on your own!