I'm from Melbourne!! I just moved to Texas a year ago. I am from the Mornington Peninsula - plenty of beautiful riding spots down there!
Try Greens Bush - plenty of trails to keep you occupied for an entire day. Mostly dirt and grass, plenty of hills. Well away from roads and the scenery is amazing. Just watch out for Kangaroos if your horse hasn't come across one before as they usually get quite a shock.
Also, try right down the end of Bungower road, Hastings. You can park just before Willow Lodge (trail riding place) and you can use their trails. Sandy trails, few nice logs to jump here and there, you can actually ride all the way to the beach if you have 4+ hours.
Lastly, try Merricks! Park near Ace High Ranch and use their trails to get all the way to Gunnamatta Beach - big huuuuuge stretch of sandy beach that is horse friendly, just watch out for the surfers getting in/out of the water. I am attaching a pic of the last time I was down there...
Other than the Mornington Peninsula, try Bunyip, Barmah and the Yarra Valley, although I don't know the trails as well in those areas, only been once or twice. Angel 3.jpg