Wouldn't we need more resources than are available on Earth?
Yes and No. But if you mean a planet, it depends on which planet. But in general we have plenty matter in the solar system and for you initial fleet it will be more than enough, even Earth resources. You can send all 10 billion of people from earth into interstellar conquest using Earth available resources, without even making a dent in its resources, but probably you will kill all animals and live on that planet during the process, but the planet itself will be perfectly intact.
In our system alone, What would we build our starting fleet out off?
Build it out of carbon. CNT's and that all.
And where would we gather the materials from?
Venus - 1.31e+20kg of carbon, atmosphere
Jupiter - about 1e24 kg of carbon
The Sun - plenty of carbon
Bonus if you can also answer for continuing to build up the fleet with resources near OTHER stars
The most common answer seems to be that there are 100 billion stars in the Milky Way on the low-end and 400 billion on the high end. But I’ve seen even higher numbers thrown around.
It has to be done that way, just because you do not have enough people, and each time to return to earth and dig for its resources to continue galaxy conqueror, is hmmm a slow process?
There will be carbon same way as here. It is everywhere in our galaxy. There will be gas giants same as here, maybe not in each system but one system per million(and we already know it is a better ratio than that) is enough.
As a note: More generally you have to read about space habitats, it will improve understanding of the problem, anyway, you may need them if you are ok with generation ships.