Yes, that "Additional 0/3" refers to cross-class skills. It means that currently you have the ability to set up to three cross-class skills and currently don't have any selected.

To get cross-class skills:

1) You need to actually have other classes with crossable skills. It looks on Lodestone like you currently have GLD at level 16 and LNC unlocked but still at level 1. (Not sure how up-to-date that is, but it's good enough for discussion purposes. I'll use those for my examples.) You unlock new classes by doing a quick quest at their class guild, just as you did at the lancer guild. Then you can equip their weapon to become that class.

Now to the part about needing crossable skills. Not all skills can be cross-classed. The first LNC skill that can be cross-classed is Feint, which you get at level 2. Each skill says on its tooltip which classes can use it. For instance, True Thrust, your first LNC skill, lists "LNC DRG" meaning only lancers and dragoons can use it. It cannot be cross-classed to anything else. On the other hand, when you get Feint, its tooltip lists "GLA PGL MRD LNC ARC MNK BRD", meaning it can be used with any of those classes (though it would take up a cross-class slot for any except lancer and dragoon). So, to see how cross-classing works, become a lancer briefly and level it to level 2 so you can get the Feint skill.

2) Once you have cross-class-able skills, you need to assign them to one of your slots. So, once you unlock Feint on LNC, switch back to GLD (i.e. re-equip your sword). As GLD, go to the Actions and Traits tab, where that "Additional 0/3" shows up. Now that you have a cross-class-able LNC skill, the LNC icon just below it should be highlighted as available. Select the LNC icon and the pane on the right will list cross-class-able LNC skills you have unlocked (which at this point will just be that Feint skill) with checkboxes by them. Checking this box indicates you want to cross-class the skill.

Once you check that check-box, the skill will show up at the bottom of the window, in the lower part of the right-hand pane. This is where it shows all your currently assigned cross-class skills for your current class. Go down to that part of the window and right-click on the icon (if keyboard/mouse) or select it and press square button (if controller) and you'll get a menu that allows you to set the skill to your hotbar. Then just choose a hotbar slot for it.

3) At this point, you'd have a GLD with the LNC skill Feint available.

Now, I used that as an example just because it's the first cross-class capable skill on a class you appear to have unlocked already. It's not necessarily the most useful for a GLD. But any others work the same way. You need the class that provides the skill leveled up enough to use it. Then you need to assign that skill to a slot on the other class that you want to give access to the skill.