It depends on what you consider inexpensive. Many people start out with a $100 Brother sewing machine from Wal-Mart that can usually handle simple sewing. Handbags and blankets are usually made of heavier materials and might make a inexpensive machine choke.
Other choices (that avoid Wal-Mart, which I like) include finding a used machine. Dealers take trade-ins on new machines and sell the old ones at a huge discount, sometimes for little more than the cost of the tune-up. Ebay, free-cycle, and relatives are other good sources for used machines. Sears sells Kenmore sewing machines that many people are happy with at below sewing machine dealer prices.
Most sewing machine dealers have mid-range machines that are around $400 and up. The advantage is in the support, almost all offer free lessons, trade-ins, service, or just having someone to ask questions. There are also on-line dealers who sell machines for less than B&M stores, the machines are usually older models and lack the support of local businesses.
Names that are good include Viking, Bernina, Pfaff, Janome, Elna. Old Singers and old Whites are also good. This isn't all inclusive, there are more good names out there.
If you can find or afford it, don't be afraid of a fancier machine. They can actually be easier to use. My computerized Viking has neat tricks like it always stops with the needle up (unless I want it to stop down) and a super easy button hole. I can switch stitch types by pressing a button. Even my 15 mo can do it, but that is another issue :LOL. A higher quality machine is also easier to use because it is reliable. When you are a new sewer this is important because you might not know if the problem is user error or machine error.
A good resource for checking out a particular model is the reviews at Sewing Pattern Review
. You have to register to view, but it is free. Good luck!