Oh wow! Big topic.
The Wikipedia article is pretty good on this:
http://en.wikipedia.org/wiki/Cave_survey
From a practical point of view you are taking measurements with Compass, Clinometer and Tape Measure (or Laser Rangefinder) between a series of stations. Stations are mostly temporary reference points just chosen for their line of sight to the next station, though at the end of a survey or at a notable junction, you sometimes install 'permanent survey stations' as points of reference. In these enlightened days you should certainly avoid the temptation to mark the cave wall by scratching it or otherwise to help your partner locate the station.
Each of these measurements defines a survey shot (or leg), which together make the centreline going through the cave.
You also record information about the dimensions & nature of the cave passage around this arbitrary centreline (typically left-right-up-down data around the station, and ideally a cross section sketch).
Back on the surface, your meticulous shot data, notes & sketches are typically entered into a dedicated survey computer program such as Survex, which does all the necessary trigonometry and trendy loop closures to give you a line drawing in Plan & Elevation, which you then supplement with your data about the nature of the cave to produce the eventual drawn survey.
The BCRA publication 'Cave Surveying' by Anthony J Day is a really good reference, and useful for symbols page alone. Apparently you can get it for the cost of ?1 postage direct from the BCRA...
http://bcra.org.uk/pub/cs/index.html?j=11
In terms of actually doing this 'for real' you'll need a set of instruments, such as the Suunto KB-14 Compass / PM-5 Clino. These are expensive. Your best bet would be trying to borrow a set of instruments, most club's sets sit gathering dust 99% of the time.
You also need a fibre tape measure, and a 'survey book' to put it all down in. 'Open Frame' tape measures are more mud resistant and reliable.
Though bringing back 'publication grade' data will require professional instruments, you can learn to survey with more basic gear - just a sighting compass and an improvised clino with a protractor + weight on a thread, perhaps? Learning to survey is perhaps 5% reading the instruments, 95% writing up your book nicely, not making blunders, miscounting stations, placing stations in good locations, communicating with your survey partner etc.
A good test of your skills is to purposefully form loops in your surveys, and see what the closure error is.
I put together some PDFs with survey books in both the 'Arrows' and 'From-To' form a few years ago. They print well with a laser printer onto A4 paper, for guillotining and rebinding as survey books:
http://union.ic.ac.uk/rcc/caving/FILES/survey_book_arrows_2010_name_survey.pdf
http://union.ic.ac.uk/rcc/caving/FILES/survey_book_fromto.pdf
And graph paper for the flipside for scale sketches:
http://union.ic.ac.uk/rcc/caving/FILES/survey_book_graph.pdf
You can also use them as templates to trace onto full blown 'plastic' waterproof notebooks with an indelible fineliner.
Over the last few years there's been significant advance in so called 'paperless' surveying, which uses a combined Laser rangefinder / clino / compass, and usually some form of handheld computer. The barriers to entry for this, in terms of equipment cost and availability, and technical ability are greater. Learning to survey by hand would put you in good stead for taking advantage of more recent advances.
Finally, of course by far the best way is to learn from someone who's currently doing this. Certainly there's lots of resurveying projects in the UK which I'm sure would welcome assistance!