An ingenious solution is the CamSextant Iphone Sextant which makes use of the iPhone’s gyro sensor. There is a CamSextant app available.
Everybody had there own preferences for the calculations. On Dutch ship the Lieuwen’s Record Tables were often used. He was a nautical school teacher. Some used the Sight Reduction Tables HO249. Very popular also were the Ageton Tables which used to be in the Bowditch Almanac but not anymore. Not even an EMP can destroy a sextant, mechanical (!) clock and the 32 pages of the Ageton Tables. The hard part today is the regular training. The accuracy of the position fix depends entirely on the accuracy of the sextant measurements. Finding a good horizon is often the trickiest part and that must be trained. You also have to know your personal error, which most people have, which must be added or subtracted to the measurements.
The quickest method these days is I suppose to use the Starpilot program installed on one of those nifty scientific calculators.