Consider a 1st level Rogue who wants to fight with two shortswords using Two-Weapon Fighting (PHB p.195) but does not have the Dual Wielder Feat.

Starting off empty-handed, is the following possible?

Round 1)

Free Interaction: Draw first shortsword

Action: Attack with first shortsword

Round 2)

Free Interaction: Draw second shortsword

Action: Attack with first shortsword

Bonus: Attack with second shortsword

Rounds 3+)

Continue happily attacking with both weapons

Or do I need to "Use an Object" on the first round to prep for the Two-Weapon Fighting?

It works

In the Player's Handbook (p. 193), the Use an Object action is described as follows:

You normally interact with an object while doing something else, such as when you draw a sword as part of an attack. When an object requires your action for its use, you take the Use an Object action. This action is also useful when you want to interact with more than one object on your turn.

As per the Other Activity on Your Turn section of the Player's Handbook (p. 190):

You can also interact with one object or feature of the environment for free, during either your move or your action. For example. you could open a door during your move as you stride toward a foe, or you could draw your weapon as part of the same action you use to attack.

If you want to interact with a second object, you need
to use your action.

You get one free object interaction per turn, which in this case could be used on your first turn to draw your first weapon, then you could attack with it. On your second turn you could then draw your second weapon as your free object interaction and engage in Two Weapon Fighting using your action and bonus action for two attacks. On the third turn, as you say you could happily continue attacking with both weapons.

On your first turn, you could instead use your free object interaction and your action (as Use an Object) to draw both of your weapons, then attack with both on your second turn, but this makes you lose one attack and there's no need to do it this way.

If you had the Dual Wielder feat, you could draw and attack with both on your first turn.

Yes, that sequence works

Player’s Handbook, page 190:

You can also interact with one object or feature of the environment for free, during either your move ar your action. For example, you could open a door during your move as you stride toward a foe, or you could draw your weapon as part of the same action you use to attack.

So yes, you could draw and attack with the first short sword on round 1, as per the rule above. Round 2, you use your action to attack with your sword and your free interaction to draw your second sword. As you are now holding a second weapon in your other hand, and you have just made an attack with a light weapon, the rule for Two-Weapon Fighting, PHB page 195, comes into play:

When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to atlack with a different light melee weapon that you're holding in the other hand.

Rules as fun

Whilst the Rules as Written would say you can only interact with one object for free either during your movement or your action, and interacting with another object would require you to use your action, the DM could rule that, on round 1, you can draw both weapons despite not having the feat.

Mike Mearls said on Twitter that “the DM is free to make a call, based on the situation” in reference to being only allowed strictly one free interaction.

Note that this may somewhat undermine the features of the Dual Wielder feat. However, as the feat does give other benefits besides being allowed to draw two weapons - benefits you would not get unless you took the feat - the DM may allow for a player to draw to weapons in one turn, simply to “avoid punishing players for that stuff by charging an action”, in Mike Mearl’s own words.

This works

So by your second turn, you have both your weapons. There is no rule against it, so I cannot give a citation.

Why not just have a sword always drawn?

Outside of big cities with strict weapon rules nobody would object.