Furniture: in Al Ain you've got Home Centre (Al Jimi Mall) and Homes R Us (Bawadi Mall and near the Diwan roundabout) and outwith Al Ain you have Ikea in Abu Dhabi and Dubai.
Home appliances: Carrefour (Al Jimi Mall, Bawadi Mall), Lulu (Al Kuwaitat, Sanaiya, Al Foah), Megamart (Al Ain Mall), Juma Al Majid (Sanaiya).
For private sellers try the following (N.B. For larger items you'd need to go hire a driver in Sanaiya who can help you shift the stuff!):
http://groups.google.com/group/nialanews - subscribe via email to nialanews [at'> gmail [dot'> com
http://groups.yahoo.com/group/alainexpats you'll need to subscribe but again useful for info
http://www.alainexpat.com has a classifieds section
Abela supermarket noticeboard near the Tawam Hospital usually has plenty of ads. You can also try the Rugby Club and the Hilton.
HI there, i'm gonna be moving to Al AIn tomorrow and will need to furnish a home pretty quickly. Can anyone give me some good places to go, where I can get good furniture with speedy delivery????