Q:

Does Rakish Audacity improve or change your Initiative bonus?

The Swashbuckler Rogue's Rakish Audacity (found in Xanathar's Guide to Everything) says:

You can give yourself a bonus to your initiative
  rolls equal to your Charisma modifier.

Is this meant to be instead of the Dexterity modifier, or on top of it?
A Swashbuckler Rogue with 3 DEX and 2 CHA rolls 10 in their initiative d20. What is their final initiative value?

A:

With Rakish Audacity, an initiative roll is d20+Dex+Cha
An initiative roll is, by definition a Dexterity check:

When combat starts, every participant makes a Dexterity check to determine their place in the initiative order.

Thus, an initiative roll will, by default always involve adding your Dexterity modifier (unless a feature tells you otherwise).
Rakish Audacity defines itself not as a replacement to that Dexterity modifier, but as a bonus to that roll:

You can give yourself a bonus to your initiative rolls equal to your Charisma modifier.

Since Rakish Audacity says that is is a bonus to that roll, it gets added on top of that normal Dexterity check.
So, for your specific example, the initiative check would be: $$10(d20) + 3(DEX) +2(CHA) = 15 $$

Jeremy Crawford has clarified via Twitter and agrees with this interpretation:

Initiative is a Dexterity check. Rakish Audacity lets you add your Cha. mod. to that check; it doesn't replace the Dex. check.

