Today there are 4 "major" brands of very high quality microscope, which are (in arbitrary order): Zeiss, Leica, Olympus, Nikon, colloqually referred to as The Big Four. In addition there are many brands which primarily rebadge Chinese-made microscopes/components and sell them under their own name, such as Euromex, Novex, Optika, Swift, Omax, Amscope, Sagitta, Bresser, Celestron, Vixen... you get the picture. This is not to mean these are all identical, as they have different levels of QC and are targeted at different segments and may be sourced at different Chinese factories. Chinese microscopes can be fine microscopes indeed, but as with anything else in life you get what you pay for. Microscopes from "the Big Four" are almost invariably pretty expensive even for entry level models (think 500-1500+ USD for entry level models), while rebadged Chinese microscopes are much more affordable. Lower price means some concessions are necessarily made.
Buying used equipment allows you get to get a much better microscope for a given amount of money. Microscopes are often sold well below intrinsic value by institutions wanting to upgrade to the latest technology. 50 years old microscopes are routinely sold at 10-20% of original list value adjusted for inflation, and if well looked after and maintained will work like a new microscope. If looking for older microscope I'd advise sticking to old quality brands like Zeiss, Leitz (Leica today), Olympus, Nikon, Reichert, and a few more. Avoid "toy" microscopes like the plague. IME they rarely have optics that justify using more than 40x magnification even if they advertise 1000x or more.
Features to look for:
- If buying used, buy from a reputable dealer, and reserve the right to return the microscope. You'll pay a premium for this, but unless you know how to service and evaluate old equipment yourself it's worth the premium.
- 160mm tube length (DIN 160 mm objectives) used to be a defacto standard for many years, allowing users to interchange microscope optics between stands of different brands. Alas, no more: Microscope makers have since switched to infinity systems which are all different and not compatible between brands. Older microscopes using the 160 mm standard allow for easier sourcing of replacement optics. Please note that Leitz used 170 mm tubes for many years before switching to 160 mm like everybody else. The 170mm Leitz optics have a good reputation nonetheless, and can be had relatively inexpensively.
- Mechanical crosstable for moving the specimen.
- Real condenser with iris diaphragm and filter holder (which will make it easy to experiment with DIY darkfield, oblique illumination and Rheinberg illumination). You will want this.
- High-eyepoint eyepiece if you wish to wear eyeglasses while working with the microscope.
- Halogen light source.
- A microscope with inclined tube and focusing which works by raising and lowering the specimen table rather than the tube. The older A-frame style went out of fashion around WW2 and has pretty bad ergonomics if you are working with live water critters because you'll need to keep the table horizontal and look vertically down the eyepiece.
Specific examples of what I would consider good entry-level microscopes: Olympus CH, Zeiss Standard, Including Zeiss Standard Junior. Other posters, more knowledgeable than me, may chime in with many more options. You don't mention a specific budget, but it's safe to assume that around 300 USD will buy you a very decent microscope which is worth much more than any factory-new microscope costing much more. In terms of optical performance, thirty to fifty years old standard quality optics IME surpass the performance of factory-new entry level optics.
And a final note:
The microscope's objective is the microscope. Everything else on the microscope is there to make sure the specimens is aligned and illuminated properly for the objective to do its work and to present you with a magnified view of the image formed by the objective. This is the reason why sticking to DIN 160 mm objectives is preferable if buying second-hand or new microscopes: it allows you to later upgrade the optics with better quality second-hand objectives, like plan achromat, fluor or even apo, without having to upgrade the rest of the microscope.