1. What does a weapon's Atk Bonus mean? I see this used but I'm not sure - is that something you add when you try to figure out if you "hit" someone, or is it something you actually add to the attack? Failed to write down where I saw it, so sorry if it's a nonsensical question...
4. On the goblin stat block, what does "+5 to hit" mean exactly?
In 5e, all actions that the DM doesn't decide succeed or fail automatically (other than spells) are resolved with a d20 roll to try to match a Difficulty Class. So, if a goblin tries to hit a PC, it rolls a d20, adds 5 to the result, and if that equals or exceeds the player character's AC, the goblin hits.
By the same token, a character sheet will have a spot for attack bonus, by weapon. Usually it's the sum of the character's Strength or Dexterity modifier (depending on the weapon) plus the Proficiency bonus for the character's level (+2 at 1st level, for instance).
Rarely, an weapon can be a magical item that adds a bonus of its own, usually +1, to that total. Players can't start with, make, or buy magic items, they're something the DM gives out as a special reward.
2. When you miss an attack due to not meeting or beating AC you just... do nothing? Cause that kind of stinks
Yes. In some other games, like 13th Age or the previous ('4e') version of D&D, at least some attacks do a little damage on a miss. In 5e, only attacks that force saving throws have an effect on a 'miss' (that is, on a successful save by the target), and, not all of them. If you have a player who doesn't want to feel like he's 'wasting' his actions due to bad dice luck, you could steer him towards a caster, and towards spells that are 'save for half' and the like.
3. What should a sorcerer or Druid be doing in their first battles? My daughter (druid) ended up just grabbing a shovel from the supply wagon and - very successfully - whacked a goblin that had run up to the wagon. Fun but I'm sure she should be... using spells? And my youngest boy is a sorcerer - what should he be doing?
5e casters have cantrips that they can use every round. If it doesn't look like you want waste a spell when it's your turn, you can always toss a cantrip.
5. I only have 3 players, so I deleted a Goblin. They barely got hurt, though, getting rid of 3 goblins. How should I balance the rest of the encounters? Just try to remove the minimum number of enemies to keep it "proportional" to 4 players?
There are encounter guidelines in the DM's Guide and in the free pdf. It's not as simple as reducing monsters proportionately to the number of missing players, but it's close. The encounter guidelines don't give very consistent results, one of many reasons why it's usually better for experienced DMs to run 5e, or to get a fair amount of experience playing with folks who already know the system before trying to DM. You learn the system, and you get to see (or at least infer) some DM tricks in action. On the WotC website, check out the Adventurers League, especially the Encounters program.
- and please point me to any learning-the-rules / first-time-DM kind of books / videos / podcasts out there. I would really, really love it if there was something TRULY step by step that I could follow, preferably written. The starter set is great, but doesn't really tell you everything you need to know
The starter set is as close at it gets. The learning curve on a game like D&D can be pretty steep. Playing in established games with an experienced DM is probably the best way to pick it up.