Coupon. Coupon. And then coupon some more

Call your local groceries and ask about their coupon policy (if they don't double I wouldn't waste your time, honestly) and pick the best to do your shopping at. Always check to see if the coupon is a better deal than an off-brand.
Off-brands will become your best friend, as well

The sites I use are :
redplum.com
coupons.com
afullcup.com
and I think that's about it.
Also, check with your local DHS office about government help programs, eg Foodstamps, WIC, there are even some help with Electricity bills, etc.
Cloth diapers and wipes (baby washcloths work wonders for wipies) also save a LOT of money, and help with pollution

If you want to get really nitty gritty, grown-ups can use cloth toilet paper too...I would wash those separate from the other clothes though lol.
Use a clothesline instead of a dryer, that helps a lot with electric bills. Wal-mart sells the retractable clotheslines that use a cuphook to hold it , I use one of those in my kitchen to dry all of our clothes. Also use cold water to wash, that helps a lot too. I do one load of laundry a day and it never builds up, that helps with stress

Also electric heat (Spaceheaters, the cheap ones from wal-mart work amazingly, one of those heat my entire house!) is a lot cheaper than propane or central heat.
Secondhand anything is superb as well, especially with baby things with as fast as they outgrow them, I honestly can't think of a single thing that I've bought my son that was new---other than diapers or wipes.
Also, if you can, plant a garden. There are even numerous things you can grow inside, because in the US fresh veggies and fruits are outrageously priced, trust me, I know lol. I really wanted to plant a garden this year but the timing of my LO arriving kind of put a hampering on that lol.
If you need any advice or help feel free to message me -- the two of us were surviving on just my husband's minimum wage, 30-hour-a-week salary, and that's only getting paid twice a month, and we were never hungry, cold or anything

It takes some planning, but you can definitely work with a budget