It's probably hard to describe any part of the Dingo Fence as being the "most interesting" given that each metre of it is almost exactly the same as the other 5,613,999 metres of it.
It is, at the end of the day, a fence. Not dissimilar to any other fence you'll ever see, with the possible exception of the fact that it's relatively high (180cm, or around 6 feet high above ground) and also runs about 30cm underground - although of course you can't see that.
Of course, it's also very long - but fences have a tendency to be very long in much of Australia, and from a visible perspective there's not a lot of difference between a 1 km long fence and a 5,614km long fence.
However if you do want to see it, the obviously place to start is, well, at the start! That would be Jandowae in Queensland, about 250km by road west of Brisbane. Jandowae has a small number of tourist items related to the fence, including some Dingo statues, and a sample section of fence in town.
Unfortunately I really have no idea how you'd get to Jandowae without a car. There is a twice-weekly train service, and daily bus services, to nearby Dalby and Chinchilla, but both of these are still about 50 km away from Jandowae.
Australia has so many brilliant places to visit - I really wouldn't be putting this one anywhere near to the top of the list, unless perhaps you happened to be in the near vicinity and looking for something to do.