Question: In a world without fossil fuels, would air travel ever really be developed?

Without fossil fuels, the most likely path that could lead to the development of a proper industrial revolution would eventually be nuclear power and renewables. This world would probably start with hydro-electric power, in which they then move on to nuclear as scientists living in the cities that build up around dams are able to discover more about nuclear physics that would allow this to be harnessed as a fuel source.

While there are other questions that could be raised, how would air travel be developed in this sort of world? Could airships remain viable in the face of electrified railroads and nuclear transport across oceans?

Would fixed wing aircraft or helicopters eventually be developed? What fuel sources might they use?

Answer: 
In Brazil "pure" ethanol (94% ethanol, 6% water) is sold at every gas station, and pretty much every new car sold can run with any proportion gasoline-ethanol mixture (they call it "flex" engines).

So, the infrastructure for a fuel that can power a combustion engine for a heavier than air aircraft does exists independently from oil since the 1970's.

The history would certainly be different, and possibly there would not be the political-economical incentives for airplanes to be what they are today, but it is quite possible they would be invented and be used for local travel in places where sugarcane is viable.
River-side mill towns existed before electricity could be used to export energy.

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> Would fixed wing aircraft or helicopters eventually be developed? What fuel sources might they use?
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sailplanes, definately.

Oowered craft, maybe for military applications, and use by the very rich. They would probably be ethanol fueled, it's hard to speculate on which type of engine they would use.