I'm a "seasoned" player that has left over XP from AD&D.  Back in those days AC was "upside down" so that a heavily armored paladin was AC 2 and an unarmored wizard was AC 10.  But at that time, wearing leather armor was AC 7, three better than unarmored.  Now I see it is AC 11+Dex when unarmored is AC 10+Dex.  I'm just venting, but I sure wish leather armor game me 3 points better on my AC as it did when I played before.  I guess I need to find a DM that will give me Very Rare Leather Armor, +2.  It would make playing my Bard more fun.  I would also really be ecstatic if I could get fighting with rapier and dagger allowed for Dex bonus with rapier and dex bonus with dagger after level 5 or so.  Those two things alone would improve my gaming experience.  Right now I have to fight with leather armor and only a rapier which generally doesn't suit me.

Bards aren't meant to be tanks. Also if you're using a rapier, just go full Dex - not only does that help you with armor but because of the rapier's finesse property you can use Dex instead of Str when determining damage.

And if you're using Variant Encumbrance, that warrior class with the plate armor is going to be encumbered unless they invest in Str, so using heavy armor is only a go-to if you're using normal encumbrance (which is boring).

Very Rare +2 Leather Armor won't generally be a thing until around level 10. There is Studded Leather (AC 12+Dex), but you will find compared to AD&D that AC and other values are lower than their previous edition counterparts.

As far as rapiers, daggers, and Dex bonuses go, both those weapons have the finesse property and so you can use your Dex for attack rolls. You can make one attack as an action with the rapier in your main hand, adding your Dex to damage (two attacks at 5th+ if you have Extra Attack) As a bonus action you can make a single attack with the dagger. You do not get to add Dex to damage unless you take the two weapon fighting style, which some Bard colleges get I think.

Also, based on some quick research. Leather gave an AC bonus of 2. Leather scale gave a bonus of 3, but that looks like it is equivalent to medium armor. Plate gives and AC bonus of 9.

In 5e, leather is +1 and plate is +8, so at least the values are consistent.

Leather armor used to be 8AC, studded leather was 7AC. But that doesn't matter, AC is balanced differently than in AD&D.

To do the fighting style it sounds like you are going for, you should get the 2 weapon fighting style and dual wielder feat. Bards don't get fighting styles by default, but the collage of swords does. The fighting style will let you add your modifier to weapon damage with your off hand and the feat will let you use the rapier and another weapon (even another rapier), get +1 to AC, and be able to draw 2 weapons at once.

Also, if your DM allows homebrew, you could make a Spiked Chain that has the same damage and stuff as a double-bladed scimitar but make it finesse, with the first attack having reach and the second attack not. It might seem like Spiked Chain is a better DB Scimitar, but because of its initially high reach an opportunity attack with it can't merged with its bonus attack, so...yeah.

Then use College of Valor. It's a pretty awesome combo. Just be aware that you can only use a bonus action once per turn so Valor's extra attack doesn't let you do SC's bonus attack twice.

You're right … it was Leather and shield that made you AC 7.  Still, I think leather deserves a +2 protection.  And that certainly isn't asking to be a tank, c'mon.

For non casters ACs for light armor and heavy armor are pretty close in 5e typically only 1 or at most two behind unless they are using a shield

Light Armor
Typically used by characters who have Dex as their main stat which will be typically be +3, getting  by level 2 they should have studded leather armor, by level 12 they are likely to have +5 Dex and +2 studded leather 
Level 1 AC = 14
Level 2 AC =  15
Level 12 AC = 19

Heavy Armor
Generally has chain mail at level 1, might get splint at 3 or 4 (when the light armour might get +4 dex) and by +1 or 2 plate by level 12
Level 1 AC = 16
Level 2 AC = 16
Level 12 AC = 19 or 20

Sure heavy armor characters have the option of using a shield but they have disadvantage with stealth checks which can cause big problems.

As a bard you also need to keep your Charisma up so you might not keep up with the rogue but going college of swords you get medium armor at level 3 so have the option of getting your AC up to 16 then even if you only have +2 dex (AC of 15 if you want to remain stealthy).

Quote from MusicScout >>

You're right … it was Leather and shield that made you AC 7.  Still, I think leather deserves a +2 protection.  And that certainly isn't asking to be a tank, c'mon.

 Did AD&D also subtract your DEX modifier from AC?

If all your saying is that it isn't as much defense compared to unarmored, when in reality both unarmored and leather have more defence than they used to, I don't know what to tell you. 

Yeah, the main benefit of light armor is that you can add your dex modifier to the AC. You can also do this for medium but with max +2. And heavy has no dex modifier applied to it. So players with classes that gain dex benefit massively from light as their levels increase, especially if they already start with strong dex. With just level 8 you could have your normal dex modifier (which should be at least +2) add with +2 (class's ability score increase on all dex) to Studded Leather's 12 AC = 16 AC. At level 19 this could be your normal dex modifier +5, so +7 means 19 AC - without stealth disadvantage and without needing high str. You end up more of a tank than the tanks. Of course this assumes you're running a dex build.

Yes, AD&D allowed AC to be adjusted for dexterity.  It is my opinion that leather should provide more protection than +1 to AC.  I'll just take it up with my DM.  I anticipate we'll come up with a solution.

The best solution would be to not expect different editions to behave mechanically similar to each other. With 5e's bounded accuracy, smaller values have bigger impact, so a +1 AC might actually be as effective as the +2 you think it should be.

Quote from MusicScout >>

Yes, AD&D allowed AC to be adjusted for dexterity.  It is my opinion that leather should provide more protection than +1 to AC.  I'll just take it up with my DM.  I anticipate we'll come up with a solution.

 
No, I mean in 5e AC is Armor AC + Dex Modifier, and classes like bard and rogue let you get 2 points to increase your score every 4 levels, so you can have a high dex modifier to increase your AC and contend with plate, with just studded. Potentially light armor can beat out both medium and plate armor after level 10.

Quote from Houligan >>

Very Rare +2 Leather Armor won't generally be a thing until around level 10. 

 If then. Getting magic items that increase AC is often pretty rare. Especially +2 armors, unless it's on the suboptimal end of the scale (like, I suppose, Leather versus Studded Leather).

Okay, scratch what I said above. Apparently studded leather only goes to 17 without magic items because of the score cap. Still though, with magic items you can go over 18. But that's also true for plate.

There is also no flat footed AC anymore, so armor AC isn't as important for that.

Quote from Rezilia >>

Okay, scratch what I said above. Apparently studded leather only goes to 17 without magic items because of the score cap. Still though, with magic items you can go over 18. But that's also true for plate.

 And heavy armor should definitely be better than light in terms of AC. You give up a whole lot to play an armor-plated tank. 

What is being glossed over is that Dexterity did used to add as much to ac as it does now. You only got a +1 bonus to ac if you had atleast 15 dexterity.

So if have 14 dex and leather armor in 5e, you'd actually have better AC than in Ad&d.
And Ad&d only had very rare ways to increase your dexterity, none from just levelling up.