Answer kinda a double-edged sword and dependent upon factors because of 'stops' and what one should (?) do at those 'stops'.
Hope you have linux ... once into the 2 series, by far easiest method as the 'march' is all command line with the scripts in admin/cli/ Server processing about the same but one doesn't have to contend with browser issues (just yet, anyway).
First hop ... should work with a copy of the 1.9.19+ (db and data directory - massive changes there). This is by far the heaviest processing on server. Matter of fact, one might want to do some tweaks to PHP and MySQL *before* making that first step - config items upwards - more time, longer time to run, larger packet size, etc. Oh, yeah, don't forget that MySQL needs to be a certain version to reach your destination (5.1.33 min.). Consider doing a MySQL upgrade even before beginning the 'trip'. Moodle 1.9 runs fine on higher versions of MySQL.
DB utf8 and utf8_collation_ci? Innodb (required later) Engine? If one did MySQL tweaks the 'waters' will be smoother.
1.9.19+ to 2.2.highest is first hop. Install the 2.2.highest via git. Using git one is ready to easily make the 'moodle march' upwards. 2.2.highest is also the first 'stop'. Has to do with last chance to get courses out of 'legacy' file system. Once a course is in 'legacy' it's 'legacy for ever'. If one doesn't care about 'legacy', continue to next hop.
Before you do, site backup. Backup becomes fall back point if next hop proves troublesome. At each stage/hop, site backup. Don't want to loose ground gained.
Next stop is at 2.3.highest via git takes just a few minutes. New Assignment added. Best, me thinks, to upgrade the Assignments at this stop. There are also quiz conversions that should be done at this stop (correcting errors/issues before moving on). Upgrade to assignments and quiz conversions require Moodle UI. Some time spent here with browser. Tip: careful not to push server when converting assignments ... do them all at one time you'd better have one hoss of a server! Disable Assignment 2.2 (the old) after upgrading them all.
Site backup.
Then it's 'smooth sailing' from there. Wash/rinse/dry/repeat kinda thing.
Next hop ... 2.3.highest to 2.4.highest via git
Next hop to 2.5.highest again via git.
Done! Finally ... well, at least you're now ready to tweak some more.
Realize the above may not follow the 'docs' but above has served me well and have experienced fewer bumps in the road as a result me thinks!
Am sure others have opinions and experiences/tips/tricks to share ... if they just will!
'spirit of sharing', Ken