what are the elements of a lavaliere mic system?
Most wireless lavalier microphones for video recording use are sold as single receiver + transmitter packages. I believe you can get receivers that take more than one input, but they are designed for AV production use (e.g. stage shows and so on) and are mains-powered and expensive. So if you want wireless you'll probably have to buy four transmitter + receiver sets. Wired microphones are certainly cheaper, but lapel mics and cables are problematic if your subjects want to move around. If they're not moving around, then maybe you don't need lapel mics.

Edit Someone has recently sent me a link to these Sony Bluetooth microphones, which look like a great low-budget option. You'd use them just like normal wireless transmitter / receiver units. 

Audio recorders usually come with built-in microphones, but microphones don't usually come with recorders, if that makes sense. Given that digital recorders are fairly cheap these days it is tempting to use four recorders worn by each of the subjects with a wired Lavallier microphone, rather than four wireless sets, but the downside to that is that you can't set levels or monitor the input, so if there are problems you won't know until you get to the edit suite. And given that lapel mics are prone to such things as clothing noise or the subject accidentally hitting them you could end up with problems.

As far as your recorder goes, you could either get a four track recorder, such as a zoom H6, or a mixer to mix the four tracks into a stereo signal which then gets sent to a stereo recorder. Recording four discrete channels gives you the advantage of being able to re-mix the audio at a later date, and to edit around overlapping dialogue, but obviously it's going to cost more.

From the recorder transferring to the computer is trivial, most record onto CF cards or such like, and the recordings can be transferred via a card reader or often via USB. Synching is also easy, even without the clapper board: editing software such as Final Cut Pro X and Adobe Premiere can synch audio tracks automatically.