The short answer: Yes, it's possible! The long answer: Finding your personal style is a life-long, ever-evolving process, so go easy on yourself and take baby steps.
Here's how I'd start: What you need is a uniform - just ONE head to toe look that makes you feel stylish and confident. A look you can toss on in a hurry and wear anywhere (okay, almost anywhere). Once you nail that look, you can start mixing things up a bit.
Need inspiration? Rip out pages of looks you like from your favorite magazines, or use Pinterest to create a personal style board. This is probably one of the best ways to figure out what you really like.
Regarding budget: You don't need to spend a lot to look good. But it helps to have a good eye - if you're shopping at Forever 21, for example, look for your more classic, timeless pieces in neutrals or black. Black always looks good, modern and usually more expensive. I'm not saying don't wear color, but it's easier to mix and match when you have a solid foundation of classic neutrals.
Ok, now for some ideas!
Uniform #1: Cardigan, cotton top, dark denim, tall flat boots
This foolproof outfit is perfect for moms on the go. Every piece is comfy and stretchy yet totally stylish. Just don't forget a couple accessories to add personality. Maybe stud earrings that your baby daughter can't tug on, and a sparkly belt to keep your cardigan in place when you're doing dishes or lugging laundry baskets around. And by the way, curvy girls can totally wear skinny jeans.