The first thing you need is a good quality stereo record deck and preamplifier, capable of playing 78s. You can play them on mono decks, but they’re not usually fitted with good quality cartridges, and even if they are, you can’t use Mono-a-Mono to clean up the recordings! You can also play them on hardware designed for modern vinyl records, but the results will be poor unless you use a special 78 rpm stylus (with the correct tip radius and shape), and can compensate for the preamplifier’s frequency response and the equalisation characteristics of the record itself.
Unless you’re on a serious budget, second-hand Goldring Lenco decks are plenty good enough for playing 78s, and because they’re relatively cheap and mechanically simple, you can modify and/or repair them to your heart’s content. Most of them also have continuously variable playback speeds, so they can cope with early recordings made at speeds other than the nominal 78 rpm.
You’ll also need a cartridge suitable for relatively heavy tracking weights (3 to 5 grams, typically), fitted with a 78 rpm compatible stylus (2.8 to 4 thousandths of an inch, depending on the record and its condition). If you’re really serious about it, you’ll need a range of stylii with different tip radii, so you can get the best out of each individual record. The Expert Stylus Company provide a near-unique service in this respect, and are very quick and helpful.
If you can afford it, connect your cartridge to a preamplifier specifically designed for the wide range of equalisation curves used in the early days of audio recording. There are a surprisingly large number of companies who still supply this kind of equipment.
Although you might need a specially designed stereo preamplifier to get the most accurate equalisation, those on a budget can make do with a graphic equaliser or software (even Mono-a-Mono’s simple bass and treble lift controls can do a pretty good job unless you’re aiming for absolute perfection). The Behringer U-Phono UFO202 is a good and economical way to get the best out of a standard record deck without having to buy a full audiophile-quality preamp.
Finally, I’d recommend buying the full version of Mono-a-Mono to get rid of the surface noise. It’s not quite as good as Cedar Audio, but what do you expect for less than 20 bucks? Oh, and don’t forget to physically clean the records before you play them. Just make sure any fluids you use don’t contain any alcohol, and if you’re dealing with really old records, try to avoid fluids altogether!