The Great Little Trading Company has good storage options. They're not that cheap, but they are robust and also are nice enough to be used after your kids have grown up a little and you want your house to look a little more adult.
It depends on how crafty you are. I bought a lovely bookcase in a second hand furniture shop. Table and Chairs from ikea, the basic wood one, I plan on painting it. The toy chest upstairs is a wicker laundry hamper from the charity shop, it's very pretty. Downstairs I have the stackable ikea recycling bins, which are cheap and keep things tidy. Storage baskets from ikea and dunelm mill. MiL loves to sew so made liners for them and the laundry hamper. She also made me a few of those jojo fabric baskets, with some basic fabric.
You can pick them up on eBay, I have a jojo search for local saved, I've picked up the bean bag chair for £5, some toys and clothes for a few quid, and a lovely patchwork topper for her bed [when she's older] for £7.50. All within 15 miles, I have a car and happy to collect.
My friend got most of her furniture of freecycle, she was happy to paint it. But someone offered her a great solid change table from mothercare, a quick clean and it looked good as new, she bought a new change mat and was well chuffed.
I didn't buy the table and chairs until DD was able to sit on one without falling off it, as we just didn't have room to have furniture sat about.
DD has a low nursing chair from the second hand shop too. It's low to the ground and has a high back, she looks diddy in it, but I can chuck her toys/jamas/laundry on and the vintage fabric is beautiful too. Cost me £10, well spent as it's lovely and solid.