Alright. Another update. We'll be starting the campaign this weekend, so things are largely finalized at this point, though I am free to make any changes I like prior to Saturday evening. Also, something I forgot to mention previously: my DM has very graciously thrown me a bone with regard to my race choice, so Dragonborn are now Proficient with History. (He thought it was thematically fitting, and considers it essentially a 'ribbon' for the race, since it's not really a 'key' skill like Perception, Insight, etc.)

There's already some good advice in this thread, but a few comments:

1) +1 on Versatile weapons if you go for Strength. The quarterstaff is versatile as a good starting point, until you get martial weapons.

Actually, since Bard starts with a Longsword, I believe sticking with that will be the superior choice. Also, due to other unrelated discussions initiated by another player, the DM has allowed the existence of a "buckler" shield, which gives +1 AC and allows the associated hand to count as "free" for most things (but not for wielding a versatile weapon in two hands, nor a two-handed weapon). I've already gotten the OK to use it along with grappling, once I get Shield proficiency, just need to confirm that it works with a spell pouch as well.

2) You're going down a Strength-based route to be good at grappling. But you don't necessarily have to do that.
<good advice snip>

This is theoretically an option, but unfortunately due to the skill/ability choices of the rest of the party, I am going to need to be the "Int skill guy" (Investigation and Arcana) in addition to my other stuff, so I really can't afford to have an Intelligence less than 12. While 14 Con and 15 Cha (pre-race) would be nice, it stretches my points so thin I will have trouble meeting the needs of the party, so I think I will probably stick with the following stats (post-race): Str 16; Dex 14; Con 12; Int 12; Wis 8; Cha 14. If I end up feeling seriously hampered by it, or the Int doesn't end up seeming as necessary as I thought, I'll ask the DM if I can re...buy? ("reroll" doesn't sound quite accurate...) at that point.

3) Your go-to spell is Vicious Mockery. It was great for Bards in 4e, and it's even better in 5e - it's one of the best cantrips in the game.
<more good advice snip>

Yeah, Vicious Mockery is a pretty sweet deal, and pairs very nicely with a grapple (since it doesn't require a free hand to cast) and makes it much harder for a grappled enemy to try to hurt me. I agree that a high Cha would help that, but...we really DO need some help in the Int department. (A new issue I hadn't noticed before: apart from the Rogue and Fighter subclasses with spells, Wizard is the only class that directly benefits from high Int, while 4 classes want Cha and 3 want Wis!)

4) A thing that people often miss - the 2nd level Jack of All Trades ability lets you add half your proficiency to all ability checks that you are not proficient in. Initiative is a Dexterity ability check - so you can apply JoAT to it.

Yeah, I was aware of this one. It's quite nice.

5) The Dissonant Whispers spell is monstrously awful, especially if you are standing next to the creature you cast it on. Shame you don't have a rogue, which would make it even more hilarious - a second sneak attack in a round is a big thing.
6) Healing Word will be very familiar to you from 4e, and works well if you're going for melee-based support. Since it's a bonus action to cast, you can still attack on the same turn.

I've certainly heard good things about both of these.

Anyway, a condensed summary of what I have chosen for my character thus far.
Leoben Nayax'ka, level 1 Dragonborn Bard
Background: Academy Graduate (Insight,* Arcana)
Stats: Str 16; Dex 14; Con 12; Int 12; Wis 8; Cha 14. HP: 9.
Proficient skills: History (racial), Athletics, Persuasion, Investigation, Insight,* Arcana
Spells Known
Cantrips: Vicious Mockery, Minor Illusion
1st level: Dissonant Whispers, Healing Word, Sleep, ???

Gear is still a little up in the air because the DM has decided to give us both the standard equipment pack and a roll for our starting wealth, so I have 140 GP (5d4*10 + 10 from backgound, rolled a 13) plus my Bard and Academy Graduate starting equipment.

*I'm really on the fence about this one. My DM is cool with backgrounds giving any reasonable skill, and "studied (Bard) magic in college" is a perfectly valid reason to get Performance instead of Insight. But Insight might have more practical use, especially since Instrument proficiency means I can do nearly everything Performance would allow, as long as I can play an instrument with it. Thoughts?

Finally, I'm not sure what to take for my fourth spell. Since I'm not pushing Cha as much as I possibly could, would I be better served taking something buff-y or that doesn't require a save? Or would Thunderwave be better, even though it does require a save? Tasha's Hideous Laughter is very highly regarded by the guides I've seen. Of course, I could always take one and then the other, but are there other/better ideas?