My vote ... if we were voting ... would be to make full 1.9 backups of courses. Stand up a fresh 3.5.highest (that's long term support) and get it minimally configured. XFer all of the 1.9 backups to a file system repo in the new server ... restore what one can from there. One will need to export all users and then import all users into the 3.5.x - watch out for admin account conflicts.
Have done 1.9.x sites to higher 3.x versions on same server before ... and if going route, would do that only by command line and using git. First 'trick' is to get the current 1.9.x under git and upgraded to the highest 1.9.x. If one cannot do that - stop ... go no further.
Full site backup at that point ... code + data directory + db dump ... that also from command line as Moodle no admin tool in any version of Moodle to do a full site backup.
Now the 'fun' begins ... map out where one is to go ... check server environment to see php version, MySQL version, php extensions. Update the component, then check in the drop down list for version of moodle anything higher than current 1.9.highest.
Along the 'march' .... and it is a 'march' ... one will have to upgrade PHP and MySQL before making next hop upwards in Moodle using git ... but you can't install those upgrades too soon or one cannot run the version of Moodle you have ... thus will also struggle getting to the next highest version of Moodle in your road map,
So briefly ... 1.9.current to 1.9.highest -> 2.1.highest -> and at this point folks will agree/disagree but I would do 2.3.highest (at that version were some major changes to Assignments ... best take care of that early ... now pick up the next hop at 2.7.highest - changes there - next 2.9.highest -> 3.1.highest ... well you should get the idea by now.
Again ... at each hop - full site backup before taking the next hop (should have mentioned that if a hop fails one has to restore the backup of the previous version that worked ... then inspect and fix what broke the upgrade o the next hop).
You'll need a lot of space to keep multiple full site backups.
BTW when/if one gets there ... Legacy file system is in affect site wide and in all courses - happened on the first hop from 1.9.highest to the 2.x.
Time frame ... well, Rome wasn't built in a day ... and neither will a 'marched' moodle ... more like 2 days ... and I mean full days ... if all goes well ... the stars and moon are aligned just right ... etc. etc.
Oh, yeah ... I didn't even ask about hardware and specs of the server where the 1.9.x resides ... other major catch 22 is just that hardware and specs. If on shared hosting ... you didn't mention where it was hosted ... don't do it ... don't even try! Higher versions of Moodle are (comparatively and IMHO) a memory hog ... does more ... needs more.
There's my 2 'sense' ... nope, spelled it right!
'spirit of sharing', Ken