A few examples of places to furnish your new home:
Het Warenhuis: - huge second-hand furniture store that will deliver to the ground-floor of your building.
Kringloop Project - Another second-hand stuff/furniture store in the centre of the city.
Marktplaats - The Dutch version of eBay/craigslist. It's another good resource for getting second-hand things for your home.
IKEA - affordable, Swedish style furniture. Nearest stores are Delft and Amsterdam. You can get to the Delft store by taking the train to Delft station, then getting the no. 80 or 81 bus. There is no Ikea online shopping for the Netherlands, you have to go and buy what you want, then give it back to them at the transportation desk, and they will deliver it to you as soon as they can. Delivery to floors above the ground floor costs extra.
Blokker - a Dutch general household store selling things such as kitchenware, cleaning equipment, and lots more for reasonable prices. They have a store on Haarlemmerstraat in Leiden.
HEMA - another Dutch general household store, good for bedding, towels, and kitchenware, amongst other things, again for reasonable prices. They have a store on Haarlemmerstraat, close to Blokker.
Zon 1,5hands Goederen - second hand store that specializes in appliances such as TVs, refrigerators, microwaves, washing machines, etc. Located on Herensingel 11 in Leiden. They deliver for a small fee.