What platform are you running Moodle on?
- Unix or Windows?
- What web server?
- What database engine?
- And what version of PHP?
How many courses do you have?
How many students?
That info will help determine the best upgrade strategy.
The upgrade from Moodle 2.1/2.2 to 3.4 is very big, so it might be better to do it in phases over several days/weeks. You can't just upgrade from 2.1/2.2 directly to 3.4 in one big step.
While you're upgrading through those intermediate versions from 2.1/2.2 to 3.4 you'll also need to upgrade your PHP version.
Along the way the Moodle theme engine has changed significantly, so you'll need a new theme (or upgrade your existing theme to the new format). Your Moodle 2.1/2.2 theme won't work in Moodle 3.4.
In 2.7 the Assignment engine changed considerably, so if you use Assignment activities in your courses then you need to migrate your student assignment data to the new format.
If you have installed any non-core Moodle plugins then you need to check if they have versions compatible with Moodle 3.4.
So it's a bit leap! Provide the above info and we can then help you determine your upgrade strategy.
You really should also make a copy of your Moodle production site and clone it onto a test server. Then you test the upgrade plan on that server before you do anything on production. And you can use it in demos to your teachers to make them aware of how much Moodle has changed and they can prepare for the changes.
Read the Release Notes (https://docs.moodle.org/dev/Releases) and New Features (https://docs.moodle.org/34/en/New_features) pages for each version to see what has changed and what you need to do to prepare your teachers and students for all the new things.