They all correct and all acceptable and they all mean the same. However, the first two can be coupled with a clause--we just started having fun/to have fun, when...(it started to rain, it was time to close the park, my mother called me in, etc.). The last one can stand alone, or it can be used to follow a plea-- Don't go, we're just starting to have fun. With a slight modification, you can also use it like the first two, but you'd have to change "we're" to "we were", ex. We were just starting to have fun when...