A safety razor is a beautifully simple tool. It holds a single double-edge blade (the same shape used for over a century) firmly in a head that exposes only a tiny sliver of metal. The razor is engineered so that, when held at the correct angle (around 30 degrees), the blade smooths away stubble without digging into the skin. It's a far cry from modern cartridge razors—and that's precisely what makes it better: less pulling, fewer ingrown hairs, and almost no razor burn when done right.
There are three fundamental types. The classic three-piece design consists of a handle with a threaded top, a base plate, and a cap that screws on. It's the most common, cheapest to produce, and easiest to clean—simply unscrew and rinse. Twist-to-open or butterfly razors have a head that opens like jaws when you twist the bottom of the handle; this is extremely convenient for loading blades and looks cool—though some models trap lather inside, so they need thorough rinsing. The third type is adjustable, featuring a dial or lever to change the blade gap. This type is not recommended for the first razor because it gives you too many variables to contend with.
When it comes to brands and styles, the sweet spot for beginners is a mild razor with a neutral, medium gap, like the Merkur 34C or the Edwin Jagger DE89. It has enough efficiency to cut the hairs without leaving your skin red and angry. You should also expect a short learning curve: the first three or four shaves will be slightly less smooth than your cartridge razor, but within a week or two, you'll notice the difference and never go back. Remember: let the weight of the razor do the work, don't press down, and only do short strokes at 30-degrees.