The fuel issue is a complex one and there are a number of factors to consider, even for a simplistic analysis. In no particular order,
1: The mass of fuel - the more fuel you carry, the more mass to be accelerated, so the more fuel you need... (and then you get into complex discussions on things like 'energy density' in your fuel system.)
2: From a practical point of view, you can only accelerate as fast as the contents of your ship can stand. As current space and military operations show, humans can tolerate multiple g-forces for short periods, but unless someone shows reasonable tolerance above 1-g then that's going to be your limit.
2b: I can not recall where, but I am sure I came across info based on long-term work in the various orbital research labs that various folks have put up over the decades which indicate that people may not do too well in the long term in low gravity. It may be that you need to maintain the 1-g for the bulk of the journey as a health factor, and that will bump up your fuel requirements. I am sure that someone will suggest rotating/centrifugal accommodation - that's going to bring up issues of long-term reliability and maintenance of moving parts.
3: However fast you can accelerate, if you intend to do anything at the far end, you have to slow down again and that's going to take fuel and be subject to the same limitations in (2).
4: Your fuel system needs to be reliable over long timescales and you want some reasonable assurance of that before you leave. This is not a trivial issue - I have worked on long-term reliability of materials, trying to predict working life over a span of a decade or two, and the whole topic is fraught with problems. So your fuel has to remain operational, all of the 'engine' mechanisms must remain operational (or repairable) over a long period and any spare matériel you carry needs a very long shelf life.
(4b: All the stuff about long-term viability applies to all the other parts of the ship.)
5: I'm sure there's other stuff I've forgotten - I wasn't a rocket-scientist.