There are two ways to get wood effect on sheet materials:
Printed Melamine, which is generally bonded to a chip-board substrate and tends to come pre-covered, it just needs a matching edge-banding... generally this tends to be reserved for manufacturing and isn't usually available/used for DIY.
Real Veneer, can be attached to anything using either hide-glue (traditional) or contact adhesive (less susceptible to moisture and heat degradation and easier to work for a large area)... thin veneers are easier to bend to awkward shapes but much more fragile, more highly figured veneers tend to be more fragile too. MDF is a popular and easy to work substrate (with the caveat that if you're machining it the dust is carcinogenic and really ought not be breathed in).
Plywood is also available with a variety of decorative hardwood veneers as a surface on one or both sides, which can be useful for paneling etc, but hiding the ply edges (and the fact that it tends to be thinner ply that's available like this) is a limitation.
At least from my point of view, it would often be easier to use a combination of solid wood and veneered sheet materials than to try to make something solely from sheet materials and veneer it all nicely... It all depends what you want to do with it as to the best way to go about it.
As to flat-pack furniture, it's so variable that it's hard to give a definite answer... My parents have a rather nice looking Ikea wardrobe that's made of chipboard with a real oak veneer, I have a much cheaper argos wardrobe that's made entirely out of solid softwood (it's somewhat smaller, and the softwood looks to be European Whitewood, which is usually considered to be structural timber not suited to making joinery or furniture... but I couldn't buy the timber for the price I paid for the wardrobe).
I'm not sure if that's actually any help and I'd certainly warn you that this isn't my area of expertise as I tend not to use sheet materials.
