Yes, you should upgrade, but you need to think about it for two reasons:
- As you say, there is the technical matter of upgrading in such a way that you preserve all your data.
- 1.9 -> 2.3 is a big jump, and all your users will need to get used to the new interface.
Regarding 2. some options are
- wait until a suitable vacation before doing the upgrade.
- create a test site, were your teachers can try out the new version of Moodle before they have to use it for real in their teaching (but you can be sure that many will not heed your warnings to look in advance of the migration).
- produce appropriate training and documentation.
Regarding 1., you basically have two options:
1a. Upgrade your exising Moodle site as explained on Upgrading. You will need to go (current) -> 1.9.19 -> 2.2.5 -> 2.3.2 (or whatever the latest versions are on each branch at the time you do the upgrade). This should upgrade your site to the new version, keeping all the existing data.
1b. Create a brand new 2.3.2 install. Leave old courses running on the 1.9 site, but whenever a new course starts, create it on the new 2.3.2 site. You should be able to transfer the course structure (resources, activities) form 1.9 to 2.x using backup and restore, but that does not handle any of the data about what the students did, just the course structure.
1b is typically preferred, because you can move each course at the time when the teacher is ready, and because your new 2.3.2 site will be 'clean'.
You also need to think about any customisation that has been done. For example, do you have a custom theme? If so, you will have to re-do that for Moodle 2.x.
If nothing else, you should probably set up a test server, with a copy of 2.3.x installed, where you can experiment, and get used to how the new version works, and what you need to do to configure it for your institution.
You should probably also plan to upgrade your 1.9.x site to 1.9.19 as soon as possible, but still proceed cautiously. Before doing any upgrading, the first thing to learn is how to take a complete backup of your Moodle site (Site backup) and restore it onto another server as a copy (Moodle migration). That ensures you can do distaster recovery. It also means that you can experiment with the upgrade on a copy, before doing it for real.