Real world switch-mode transformer design is not straight forward, especially not as simple as engineering courses in college made it seem. Many companies that do a lot of power supply design will have Magnetics Engineers - engineers with a EE or Physics background who do nothing but do the design and design testing of inductors and transformers.
The best design guide and reference for transformers can be found in a series of books written by Colonel William T. McLyman such as Transformer and Inductor Design Handbook which has been published in several editions (I've tripped across scanned pdf versions of older editions online). It's full of step-by-step examples. However, the book is around $130 new.
Otherwise, go to coremaster.com and look at their application note AN109 which was written by McLyman for a push-pull application. A lot of what goes on in his approach are calculations to make sure the core size will support the wattage, the core window will be large enough to hold the winding needed, and what the approximate temperature rise will be. Just using this app note without addition examples you will miss some details (i.e. the skin effect calculated is as-if the excition is sine-wave, but reality will be switching with harmonics, and there is a unexplained SQRT2 factor applied to wattage in the secondary calcualtions that arises from the example being center-tapped that may be different if you have a 4-diode bridge). The fact that the Coremaster example is for a toroid will not change the design method, and following it and even making minor mistakes will result in a transformer that will work.
You didn't indicate what material your cores are, but you can assume a safe flux density of 0.06 Telsa for most ferrites (Note - Coremaster makes exotic nanocrystalline cores which support a huge flux density with a potential downside of huge inductance). A few quick calculations indicates that your ETD34 core is insufficient for 300 watts (doesn't meet the Kg core geometry miminum needed). Going to a much larger ETD54 core will meet the Kg minimum for a reasonable regulation (Alpha of slightly over 1%), but then you find your window factor is almost 0.6 (not enough room for the wire), so if you decrease your wire size (and increase the losses in the copper) you can make it fit, but It'll have a temp rise of 40 to 50C.
As you can see, transformer design can be a very iterative process. To complicate it, many different magnetics manufacturers use different units for their magnetic materials. Some follow metric units, some follow English units, some mix both, some publish theirs with a slightly different variable symbol where the units (not given) are in mm instead of cm just to be different from a competitor. It can be a daunting task just to make sense of it all.
Of course, I've just commented on the design of the transformer. Since you want to do push-pull (a very straight-forward method and very good choice for the power level your doing) you also have to consider the complexities of a switching circuit, and things like 'flux-walk', 'current mode' Vs 'voltage mode', and the extra parasitics you will have from a transformer that is stepping-up to a high output voltage.
Now that I've discouraged you, just go to Linear Tech, look at page 1 of Application Note 73, have a laugh, and proceed.
Good Luck!
