I do not want to say that second hand microscopes are generally worse. Quite the contrary. Many of them offer much more value than new ones of a possibly higher price. For the beginner, second hand microscopes are problematic, however. Many of the possible problems can not be seen and it is necessary to first test the microscope. Some hospitals may occasionally sell used microscopes. Be aware that these devices are sometimes equipped with specialized optics, which are not suitable for the hobby microscopist, but which can be quite expensive.
Do not buy specialized microscopes such as inverted microscopes, metallurgical or polarizing microscopes, unless you have specific needs for them. Again, if you buy second hand microscopes, you may not always know the difference, because the person selling the microscopes may not know the difference (unless you buy from a microscope dealer). If you buy microscopes for children, then do not buy ones that have only a mirror instead of a lamp. Some children may point them to the sun and do damage to their eyes. Mirrors also do not provide enough light intensity for photography, unless you position a lamp right in front of the mirror. Also do not buy historical microscopes, unless you are a collector of old microscopes. Historical microscopes can be quite expensive and nice looking, and it would be a pity to use them for routine observations. Old microscopes may also not have the optical quality, as there could be the problem of fungal growth on the eyepieces and objectives.
Last, I would discourage the buying of toy microscopes. These are microscopes that are sold in a colorful cardboard and Styrofoam box together with a wide range of different accessories. There is a general agreement among enthusiast microscopists and teachers that these microscopes should not be bought. First, they are not as cheap as one may think and for a little more money one can already obtain a microscope with substantially better optics and more long-term value. Toy microscopes are often difficult to focus becasue they often do not have a coarse and fine-focus knob. Do not forget, that the tolerances of the mechanics has to be extremely narrow. Plastic gears simply can’t keep up with metal gears. They do not have standardized objectives and the resolution of the picture is low. Often the magnification is also advertised as unrealistically high (750x or even 1000x). The low light intensity (battery operated or mirror) makes it difficult to see the specimens properly. If money is indeed an issue, then it’s better to get a simple but solid stereo microscope. They are more fun to use. In my opinion, children need stable and solid devices that produce a sharp, contrasty and bright image. Kids are demanding these days. The images that the microscope produces has to compete with the strong visual impressions from television, the Internet and magazines. A low-contrast, washed-out, dark picture produced by a toy microscope will not captivate the children for an extended time.