What digital cameras allow me take picture of small objects at close distance?

For Point & Shoot type of cameras, you'll need to look for "macro" (Nikon, Canon usually can do 1-2cm, which means you can be that close to your subject). For SLRs, you'll need to buy a macro lens. A good website to find out these specs is www.dpreview.com

My suggestion : Casio Exilim

The feature you need is called "macro". Most decent digital cameras are capable of macro photography, and have special settings for it.

It is also important to have a tripod to hold the camera steady.

Pretty much all of them. Post another question telling us what you want to photograph and how much you want to spend and we can give you a more meaningful answer. 搂