So I'm guessing that Dragon's Dream makes it easier for you to incinerate enemies with Igni like how the developer showed in the gameplay trailer? In that case, the damage would depend on how far you leveled up your Igni sign correct?

I just started the game so I'm in Chapter 1, so I haven't put any points into my signs yet. I got ingredients to either make a Dancing Star bomb or a Dragon's Dream bomb. Which bomb would be better? Would Dragon's Dream do more damage at the cost of having to use vigor to cast Igni?

I'm not up on the stats for each right this sec, but also keep in mind you can ignite Dragon's Dream bombs with Dancing Star bombs as well (and maybe others), getting around the vigor cost. And Dragon's Dream bombs can chain react, according to their description.

There is a talent that will give bombs 30% more damage. I believe that it is the first option on the bottom group of talents.