an addendum to my earlier reply.
Sydney: only three summits are within easy reach of the metro area, SY-001, SY-002 and IL-018 to the south. Compton's offer to assist in the Sydney area is worth considering, he knows where everything is and knows a few other things too, like the CT summits that are worth more points and just a bit further to the west. Ed's suggestion of HU-093 is good too, depends on your time available to spend a few hours heading that way.
En route Sydney - Melbourne. Recommend stopping off at IL-001 Mt Gibraltar which overlooks the towns of Mittagong and Bowral and is a drive-up. Another summit nearby requiring a 20 min walk is Mt Alexandra, VK2/IL-005, to the north of the Mittagong town centre, steep final climb to the car park in your rental car but feasible. Local name is Katoomba Lookout. overlooks the Hume Highway.
Further south, divert to Canberra (VK1) just after Goulburn. This gives you access to Mt Ainslie VK1/ac-040, Black Mt Vk1/ac-042 and Mt Stromlo VK1/ac-043, all drive ups. See comments by Andrew VK1AD re alerts and letting the locals know - there are plenty but they need notice. If you time it right you could meet all the VK1 SOTA guys for a meal.
On the way out of VK1 back to the Hume Highway at Yass, Mt Mundoonen, VK2/ST-053 just 5 km east of the junction of Barton and Hume Hwy, right next to the highway and you can drive up most of the way. Stop when you worry about your vehicle damage agreement. Some NDB antennas and other masts at the top. Even if you walk all the way it's only 20 mins or so.
Heading south-west along the Hume, pass through Gundagai (stop 5km before the city to see the dog on the tucker box - famous icon) then continue towards Albury. Some nearby summits along the way, just south of Albury you go over the border into Victoria (VK3) and Huon Hill and Mt Big Ben are both accessible and quite close to Wodonga. After that my knowledge peters out as I haven't activated much in that area. Others have more local knowledge than me.
Your planned trip from Sydney to Melbourne seems to be in the week of the 10th to 14th July, weekday activity is definitely lower than on weekends, but if you alert beforehand, you will make enough contacts to qualify the summits.
Hope that helps.
vk1da/2uh at Yass