Good question! The short answer: LOTS of things. Improving a jet engine means making it more efficient (so that it uses less fuel for the same task and emits fewer harmful gases), making it safer, making it quieter, making it easier to maintain, making it cheaper to manufacture.
A jet engine is incredibly complicated, but it can be broken down into smaller parts that have their special functions: for example, the fan, the compressor, the combustor, the turbine.
All of these parts can be broken down even more, like blades, pipes, valves, shafts, disks, etc.
Your suggestions are spot on; the shape of the fan blades can be changed to make the air flow smoother, the material of the hot turbine blades can be changed to withstand hotter temperatures, the fuel spray nozzle can be changed to better mix the fuel with the air (so that it combusts quicker and produces less harmful gases), etc.
There are many such things that can be done to improve the jet engine, and each of them takes a long time and lots of determined people to research.