Cheap children's bedroom furniture can be acquired in two ways: You can buy it used, or you can find a really good sale on new furniture. Of course, you can always find free furniture by asking friends and family for hand me downs, which is the cheapest way of all.
Used furniture is almost always cheaper then new, even when the new items are found on sale. Good places to look are the Goodwill, Salvation Army and other non-profit thrift stores. Since their items are donated and the money collected goes to the less fortunate, items are sold at rock-bottom prices. If you can't find what you need, other good places to look include classified listings, garage sales, online swap groups, second hand children's stores and semi-annual consignment sales.
If you prefer new furniture, you're not out of luck. Good prices on new items are difficult to come by, but deals can be found with patience and persistence. Start by looking at weekly sale ads in the newspaper, where stores list their best prices on regular inventory. You can also visit local furniture stores to see if they have a clearance section for closeout, scratch and dent and returned items.