I know they are important as a primary source for history of WW1, but they have just been kept in an old suitcase by my family throughout the generations. I am seeking advice from historians as to what i should do/ how i should go about possibly re-writing or photographing each page and posting them somewhere.
You're right, these diaries are valuable, and worth preserving. Your family history that goes with the diaries is just as valuable - it puts them in context.
The Australian War Memorial is well-established, well-funded, and specialises in this material. They have an online donation form here https://www.awm.gov.au/collection/donations/, but if you're not sure whether you want to part with the diaries, phone the War Memorial and ask to speak to a curator about what you have. They will be able to advise you on how to keep them safe and intact, and options for making the contents available.
Many museums have now established efforts to photograph documents in their collection, and to get them transcribed by crowdsourcing online. You and other members of your family would have a chance to help with this effort.
Talk to the War Memorial before trying to photograph the diaries. They will have valuable advice on handling possibly fragile works on paper, and might want to do the photography themselves, as they will have internal standards for image quality and so on.
(Source: Collection Information specialist for the Powerhouse Museum, Sydney, for 18 years.)
You could try contacting the Australian War Memorial in Canberra, they'd probably be pretty interested.
Depending on where you are in Australia, some good options would be the Australian War Memorial or state library. Some of the larger universities might be interested too
I have a related question. My father kept a diary while he was a POW in WWII in Germany. There is nothing of great historical significance but it might be of interest to historians of that era as a primary source. Does anyone have a suggestion of who might be interested? (Actual diary here: http://imgur.com/a/lCE6W#0 if anyone is interested).
The National Archives is also worth contacting. They also have specialists in preservation and digitistion of old documents as well as a photography division.
maybe x-post this to /r/TheGreatWarChannel and ask Flo and Indi. they'd be very interested in the contents at least.
As other commenters have said, your first step should be the Australian War Memorial and then possibly the State Library of New South Wales or NSW State Archives for advice on where and how to go further. I say New South Wales since you referenced 20th Battalion AIF which was raised in Liverpool, in western Sydney.
I'd absolutely get in touch with the Australian War Memorial.
I live in Canberra so if there's any practical "on the ground" way I can help please let me know.
I would somehow make copies of them without potential for damage, then take the advice already given and contact a museum/war memorial.
After that, I'd create a subreddit and submit diary entries on a scheduled basis. A sub comes to mind that does this already with a Civil War era diary. It's quite fun to read, sometimes anxiously awaiting a new entry. I'll try to track down a link for it.