If you know how to crochet in the round you can easily make a bag without sewing pieces together. To make a basic bag shape you have two options: you can make a magic circle to start a round bag, or you can work off of a starting chain to make a more rectangular or oval bag.
To make a round bag, start with a magic circle. If you increase evenly all the way around you will have a circular bag. Increaseing on two opposite sides of the circle as you work will make an oval bag. Once you have your base as big around as you'd like your bag, you can start working the sides by stitching around and around without increases. You can work your first row of the sides in just the back loops of your base to give a more clear line of definition between the bottom and the sides of the bag.
Making a bag from a starting chain will give you the option of a bag shaped like an envelope without a flat bottom, an oval bag, or a more rectangular bag. To begin with this method, make a chain.
Determining the length of your chain without a pattern can be tricky. You need to keep in mind that your stitches will be working around this chain the way a braided rug is wrapped. This means that for every row wider you want the base of your bag, it will get proportionately longer as well. So make your starting chain one stitch shorter than the ideal length of your bag for every row of width you want at the bottom of the bag. If you wanted a bag that was ten rows wide (five on each side of the chain) you would leave your chain five stitches shorter than your intended length to compensate.
Now that you have your chain, its time to decide what shape you want your bag.
To make a bag like an envelope, you wont need a large number of increases so make your chain almost as long as you want the bag's bottom to be. In the first stitch of your chain, make two single crochet stitches. Single crochet the rest of your chain except the last stitch. Make two single crochets in the last stitch. Now you've reached the end of your chain and instead of turning your work and going back over the row you just made, work down the opposite side of the chain. Reverse the pattern you performed earlier with two single crochets in the first stitch of the new side, single crochet to the far end, and two single crochets in the last stitch.
At this point you could start working rounds without increases to make your envelope bag. If you were to add increases to the "points" of the corners your bag is starting to form, you would start to create a flat, rectangular bottomed bag. Adding several increases across the ends of your bag would give more of an oval shape.
You can make bags from these bases with any stitch pattern you please, you just need to calculate your starting rows to accomdate the stitch you want to use. If a stitch has a multiple of 4, begin your magic circle or base chain with the correct number of stitches. You can also work the bottoms of your bags in a simple stitch like single crochet and the sides in something more decorative.
There are thousands of bag patterns available online. Looking a few of them over thoroughly should give you an understanding of how bags are constructed. From there you can make a bag in any shape you want.