Audacity (see Resources) is a free music editing program that can be used to record music from multiple sources and then edit it. With Audacity, you simply open a new track and hit record while playing your record. The record player has to be connected to the computer through either a phono preamp or through a receiver with a built-in phono preamp and an audio out in order for the levels to be high enough to record. After recording the music, use the Add Track function under the Program menu to add track marks for each song, then click "Export as MP3" under the File menu.
Use Spin It Again ($34.95, 2010 prices; see Resources) to not only record your audio but to clean up pops and hisses. The Cleaning and Effect Processing wizard is available for both songs and the entire music file and can be applied globally, instead of to short sections of music. Spin It Again also has a handy feature that automatically recognizes the silent spaces between songs on a vinyl record and inserts new tracks and has support for standard sound card hookups for old turntables and USB hookups for modern USB turntables.
Use the ION TT USB ($119, 2010 prices; see References) to bypass an audio receiver and record your albums directly to your computer via a simple USB connection. The ION comes with a program called EZ Vinyl Converter, which lets you record the audio to your computer, insert track marks for each song and then export the files as MP3s. You can even insert the track marks on the fly while recording from the EZ Vinyl Converter control panel.