Hello everyone,
I am a new administrator of an online school and it is currently running Moodle version 1.9.14. The PHP version is 5.6.40/
I am wondering if it is possible to update the site to a newer version of Moodle, and if so how could I go about this? I am new to all of this and have absolutely no idea how to proceed.
Official docs:
https://docs.moodle.org/403/en/Upgrading
https://docs.moodle.org/403/en/Upgrading
that's just a start!
Now some questions for you:
1. How are you hosted? Local or remotely with a hosting provder
2. what operating system is your server
3. do you 'suffer' from 'clas' ... 'command line avoidance syndrome'? 
Well start with those.
Also, expect a PM from me. Please respond there!
'SoS', Ken
New administrator, welcome! Your Moodle can not be called new though, 1.9.14 was released in October 2011. Since then about 25 major versions were released! See https://moodledev.io/general/releases.
Rightly you ask about a "newer version of Moodle". If your people are currently happy with version 1.9 there can't be a big hurry to catapult to the latest and the shiniest, except for the administrator, a 1.9.14 must be having hundreds of documented security holes.
But for those coming from Moodle 1.x there is some bad news. The transition to 2.x broke so many things, it is advisable to
rebuild the site - either through course backups and restore or create
courses from scratch making use of the numerous new features added since
then.
Anyway, if you want to try the long march, you can plan your route from this chart: http://www.syndrega.ch/blog/#php-and-dbms-compatibility-of-major-moodle-releases, which I happen to maintain BTW. For each upgrade there is a document page like https://docs.moodle.org/19/en/Upgrading. If you have technical questions these are the kind of technical information you need to provide: https://moodle.org/mod/forum/discuss.php?d=356184#p1436759.
If the OP's server is Windows, even more 'fun' is in store.
PHP on Windows was 32bit ... even on Windows servers that were 64bit. When courses reached the size of 4Gig's, courses couldn't be backed up.
So if Windows and PHP is 32 bit, OP is looking at a migration to Linux first - with dummy downed PHP and MySQL/MariaDB just to do the migration and view the site.
Next is the Assignment module ... which had a major change exactly at version 2.2 of moodle. In the site admin menu of a 2.2 moodle there appeared an 'Assignment Upgrade Helper' item. All assignments had to be converted to the new 'assign'. And, the old assignment 2.2 at that time was a module that was hidden but remained in core moodle until 4.3.
So even backing up courses from the 1.9.x in hopes of using on a freshly stood up 4.1 or higher is iffy.
Best might be to stand up a server with version desired, then mimick what is seen on the 1.9 in the 4.1 or higher - rebuilding the site ... one course at a time.
OR ... forget the whole thing and just do 4.1 or higher on a Linux server and build new courses ... which in reality might be the best - considering information in the Globe doubles every 6 years!
Kinda wonder how useful courses would be that were developed/used in 2011 now.
Anyhoo, Molly ... aren't you glad you asked! 
'SoS', Ken