Would it be possible to have a planet made up of about 60-70% water with no landmasses larger than Australia? I would like this planet to be as earth like as possible, but I'm open to changing aspects of it, such as rock composition, size of tectonic plates, etc.
Yes it is possible to have a planet like that. But it will be different from Earth in a couple crucial ways.
You can have it in the habitable zone. You can have it near-Earth sized. You can have the same elemental composition. But you cannot have it tectonically active. Tectonic activity comes with high probability that the cratons would collide and merge to form larger landmasses.
The most primitive continents on Earth after the coming of water, were the size of large Japanese/Indonesian islands. There were a lot of them and they were moving around rather fast (due to tectonic plates motion). Wikipedia article says:
Although a process similar to present-day plate tectonics did occur, this would have gone faster too. It is likely that during the Hadean and Archean, subduction zones were more common, and therefore tectonic plates were smaller.
EDIT TO ADD
There is no such thing as adverse effects. What is beneficial and what is adverse is a completely subjective perspective. You may have a planet with a molten core and static crust. But you would have to heavily tweak several geologic features of your planet (especially the composition of the core and the thickness of the crust, along with the percentage of water and land). These changes are in their own right another subject of discussion.
While Youstay Igo is essentially he correct that does sort of miss the point that for millions of years the planet can have the configuration you need.
While earth can have the layout you want you can make it more likely by a few simple adjustments. Essentially all you need is for there to be a little more water on the planet, or for the oceans to be a bit shallower, and then to split up the continents a bit more.
So far as we know there are no hard rules restricting the amount of water on a planet so you can adjust that along with the landmass positions to get your desired result for a few hundred thousand or even millions of years.
Eventually your landmasses will merge, but then they will split apart again. You have plenty of windows to visit the planet in where you get the layout you want.
Early earth, pre-Pangaea and even pre-Laurasia, was kinda what you are suggesting here... so ya - it's not only possible - it has happened on our planet. Mind you, plant life had yet to alter our atmosphere to nitrogen/oxygen and the air was outright toxic to what we consider life now.
Major note is: you either need a relatively young planet with tectonic plate movement, or one without these plates (no plates will cause a very shallow ocean with a few islands poking out from volcanoes). Young tectonic plates works as well, though a few million years will see the rise of mountains as the plates crash.
I'd say it's easiest to go with no plates for this setup. Volcanoes will be the majority of your landmass... and they can grow really large (think Olympus Mons on Mars). That said, I'd have no idea what impacts to larger climate that would have. It'd be really weird to be in the middle of the ocean... and have it be 16 feet deep.
Added - remember you will not get island chains very easily in this setup. Island chains are created by a volcano as the plate slowly drifts over it...first island created by volcano, continent drifts and relocates the volcano island from the volcano,same volcano creates another island beside it and so on. Instead, you will likely have single landmasses (islands) that have a central volcano / mountain in the middle and overtly lush green lands down from there, desert for a bit, then ocean (use the large island of hawaii as an example). Volcanos should be decently regular as a planet without plates only has volcanic activity to release energy.
As an adverse effect..there will be a risk that the increased volcanic activity will poison the atmosphere with sulphur and other gasses that are not life friendly.
A second effect will be drastically varying life forms as each island is badly isolated. Each island could have its own very unique plant and animal life. Possible that introducing a creature from one island to another would collapse the entire islands ecosystem....tad bit more fragile.