1. Spell as bonus action, Attack as action
You could do this. Certain (very few!) spells have a casting time of "1 bonus action." As long as you meet all of the other requirements for casting such a spell, there's no reason you can't Attack as your action and cast your spell as a bonus action. (And those can happen in either order, too.)
Unfortunately, only one spell that's accessible to a level 1 Bard has that casting time: Healing Word. I imagine you're looking for more spell+sword action than just that....
2. Spell as action, Attack as bonus action
First off, everyone can dual-wield. It's really badly laid out in the PHB, but on p.195 you'll see the Two-Weapon Fighting section--this applies to all characters. (It's easy not to catch this, as by this point in the PHB you've passed the Fighter's two-weapon style (p.72) and the dual wielder feat (p.165). Like I said, the organization's just horrible sometimes.)
Unfortunately, dual-wielding won't help you here: to attack as a bonus action with your second weapon you must have taken the Attack action earlier in your turn.
3. But I really want to cast a(ny) spell and swing a sword!
You are not alone. Since the delineation of the Magic-User and Fighter classes in original D&D players have always wanted to do this, and designers have always made it really hard. In 5e it's largely the action-economy that makes it hard.
How hard is it? Let's take a look at some of the ways to be a spell-sword:
- Your simplest option is already in your class: choose College of Valor, wait until 14th level. That gets you the Battle Magic class feature: "when you use your action to cast a Bard spell, you can make one weapon attack as a bonus action."
If that's too slow for you, here are some other options:
Take two levels of Fighter, any archetype, gaining the Action Surge class feature. This'll get you one extra action on your turn: you can cast a spell as one (standard) action and Attack as one (standard) action. But you can only do this once per rest.
Or take three levels of Sorcerer for Metamagic (Quickened Spell). With your sorcery points you can convert the casting time of a spell from 1 action to 1 bonus action. But you've only got three points, and need to keep taking Sorcerer levels if you want to accrue more points.
Next up is to take three levels of Cleric, gaining access to the spell Spiritual Weapon. Cast that and, for one minute, you've got the ability to move the weapon up to 20' and execute its attack, all as a bonus action. This leaves your standard action with which to cast a(nother) spell. Perhaps not quite the "sword-and-spell" style you're going for, but at least it's "spell-cum-sword-and-spell". (You could also wait until Bard 10 and take Spiritual Weapon as your Magical Secret, as @Escroteitor points out, to the same effect.)
What you're imagining sounds like what you get if you take seven levels of Fighter (Eldritch Knight archetype) to gain the War Magic class feature: "when you use your action to cast a cantrip, you can make one weapon attack as a bonus action." (PHB p.75, emphasis mine) With this option you can spam away your damage-cantrips, swing for the fences with your sword, singing away merrily the whole time, and all rather quicker than if you'd waited until Bard 14.