I'm new to D&D and just started making my first character, she's a wood elf druid.
On the skill sheet there's a section on "spellcasting ability". I've tried looking elsewhere to find out how to calculate what it'd be but can't find any way to do so. How do I find out what my spellcasting ability is?

Each spellcasting class will have a section in the class description where it describes what ability score is used for spell attacks and for spell save DCs. For a Druid, it's Wisdom (page 66 of the Player's Handbook):

Spellcasting Ability

Wisdom is your spellcasting ability for your druid spells, since your magic draws upon your devotion and attunement to nature. You use your Wisdom whenever a spell refers to your spellcasting ability. In addition, you use your Wisdom modifier when setting the saving throw DC for a druid spell you cast and when making an attack roll with one.

Note that it says "for your druid spells" in this quote. If you have the ability to cast spells from more than one source (such as multiclassing, a racial feature or a feat), you may have more than one spell casting ability. Each rule that gives you spell casting will tell you what ability to use for the spells it gives you.

In the Player's Handbook, in the class description

The rules for a character's spellcasting ability always appear in the class description.

For the druid, that appears in the Player's Handbook, p. 66, under the heading Spellcasting, has a subheading, Spellcasting Ability, which describes the druid's spellcasting ability.

The druid's spellcasting ability is Wisdom.

Wisdom

Wisdom is the druid's spellcasting ability. The details are located on p. 66, Paragraph 6 of the PHB (Player's Handbook) under the Spellcasting header.

Your DC will be 8 + your proficiency bonus (which is +2 for a level one character) + your Wisdom modifier. Let's say you have a Wisdom score of 14; that means your modifier is +2. That puts your total spell save DC at 12 (8+2+2).

Your spell attack modifier is equal to your Wisdom modifier + your proficiency bonus, which equals +4 in this case.

This means that if you cast a spell that requires a spell attack roll, you will roll a d20 and add 4 to it.

Here is a link to a video about casting spells in Dungeons & Dragons that you might find useful.

If you don't have the PHB, here is a link to the basic rules; you should find everything you need under the druid tab in classes.