There are many acceptable ways to digitize recordings, ranging from incredibly sophisticated (and expensive) audio engineering strategies employed by professional recording studios to all-in-one USB turntables for hobbyists. I've tried to compile some basic information below if you're trying to get started, but "Guidelines on the Production and Preservation of Digital Audio" (IASA TC-04) and Casey and Gordon's Sound Directions can offer much more information about accepted archival practices.

Your turntable is your primary point of contact with the recording. If you're investing in a new system, I recommend against purchasing a USB turntable with preamp and ADC included. Look for a turntable that will allow you to adjust speed and tracking force, and choose your cartridge. Although 78rpm discs are monophonic, a stereo cartridge will double the information gathered and allow you to restore the digital audio more easily. Having several styli of different sizes and shapes is the best investment in producing high quality transfers. These can be purchased here. If you can only afford one, a 2.8 truncated elliptical stylus is a good versatile choice. More technical information on this topic can be found in Powell's "Audiophile's Technical Guide to 78 RPM...".

Unfortunately I can't recommend a consumer system for transferring cylinder records, but the Phonograph Makers' Pages describe a variety of more or less homebrewed solutions if you're crafty.

Phono Stage / Phono Preamplifier
The phono stage amplifies the electric signal produced by the turntable's cartridge to "line level", and may offer equalization curves that emulate the response curves of original playback equipment. The KAB VSP MK2 is full-featured and reasonably priced, while the Rek-O-Kut Audiophile Archival Preamp offers more limited features at a bargain. If already have an LP preamp, you can re-equalize your transfers with Brian Davies' Equalizer application. Alternately, if you're using Audacity, you can use the "78RPM EQ Curve Generator" plugin to equalize from a flat transfer (given proper turnover and rolloff values, some of which can be found here).

Analog-to-Digital Converter (ADC)
The ADC receives the equalized line level signal from the phono stage, and records the amplitude and frequency of the signal many thousands of times per second (this is the 'sample rate' referred to in digital files). The combination of each of these points of data comes together to create a waveform that can be captured by your computer. The Benchmark ADC1 is very popular in professional applications such as archives and recording studios. I have had good results at home with my Tascam US-144MKII unit.

Computer Workstation
This can be your home desktop or laptop computer, Mac, Linux or Windows. Doesn't matter. You will not need a high performance gaming machine with hundreds of megabytes of video RAM, but you do need a computer with a fast enough CPU and enough RAM to keep pace with your ADC module.
  • Sound Card
While many ADC units designed for home/hobby use will have USB or firewire connections, many professionals opt for a stand-alone PCI audio card, such as Lynx Studio AES16 (AES/EBU compatible) or AVID Audiophile 192K (S/PDIF compatible). I currently import via USB from my Tascam ADC, but will probably upgrade to an internal sound card eventually to take advantage of the available S/PDIF output.
  • Digital Audio Workstation (DAW)
Your computer will need Digital Audio Workstation (DAW) software to capture the signal from the ADC. Your DAW software does not need to be fancy or expensive. In fact, importing digital audio is a fairly simple task. Your choice of DAW should consider your budget, your workflows, and learning curve. Audacity is a good open source option available for Windows, Mac OS and Linux. Goldwave is another good option at $50, with many advanced features and good denoise capability.

Digital Files
Current practice in archives is to capture as much information as possible when digitizing objects. This means that records are saved as uncompressed wave format digital audio (.wav), or it's metadata enabled cousin Broadcast Wave Format (.bwf). Audio should be captured with a sample rate of 96 kHz at 24 bit depth.

These lossless files can be expensive to distribute, but fortunately archives best practices also recommend the creation of an 'access' copy which may be compressed and (unlike archival wave-format files), digitally restored to remove unwanted hiss, clicks and pops.

Obsessive audiophiles may opt to encode access files using the lossless FLAC codec. FLAC removes only information outside of the human audible spectrum and information within the spectrum that would be too quiet for us to hear.

Lossy audio compression formats such as MP3 and AAC technically remove information that may be audible in some way, but a high quality transfer into a high bitrate mp3 file works well for most applications. Erring on the side of quality, I typically encode at 192 kbps.

For further discussion on this topic, read the 2009 report by the technical committee of the Association of Recorded Sound Collections, or chapter 3 of Casey and Gordon's Sound Directions. For the theory behind the recommendations, see this.

Storage and Handling
Always store disc records upright on their sides (never stack them). Dividers can partition long shelves and minimize physical stresses. Hold a disc by the outer edges. Avoid touching the grooves.

Most archives use vacuum machines like Keith Monks or VPI to clean their records, but I believe you can get nearly the same results with a soft-bristled brush, Disc Doctor cleaner and filtered water. Most 78rpm disc labels are waterproof, but if you're concerned about this, or washing lacquer discs, the Groovmaster label saver will keep water away from the label while washing. When the disc is dry, place it in a new clean sleeve (I use Disc-O-File).

Read the Library of Congress' "Cylinder, disc and tape care" for more information.