I'd like to beat India the most, from a personal perspective. England and the Ashes come a clear second, though, so don't worry.
I'd also like us not to lose to South Africa - Graeme Smith will be utterly insufferable if they beat us.
On the contrary, I don't think that India brings out the best in Australia - more like the worst. For instance, I've noticed that, all too frequently, we're beset by some kind of notable problem when we play them. In 1997/98, we had a third-rate bowling lineup. In 2000/01, we were missing Brett Lee (when he was still regarded as pace bowling's new sensation) and Shane Warne was, at least psycholgically, underbaked. In 2003/04, we were missing Darren Lehmann (a very good player of spin), along with McGrath and Warne. Even when we're
not missing players, an aspect of our game seems to drop off: in 2000/01, we looked like idiots against some pretty good spin bowling and in 2003/04 and 2007/08 our catching was laughable. The only times when we weren't beset by serious issues or lapses in key areas would be both 1999/00 and 2004/05 (when India endured key injuries instead).
If you ask me, I reckon that Adam Gilchrist had at least something to do with this. He averaged 28 against India overall and also dropped several catches and missed at least a few stumpings during this series just gone. Against everyone else, he averaged 40+, which may go some way to explaining our success rate against everyone else.
I agree with the rest of your sentiment, though. India seem to up the tempo when they play us, at the same time when we usually seem to drop off, leading to some competitive series (2000/01, 2003/04 and 2007/08). It's probably good for the game...that being said, I'd still like to beat 'em more than anyone else.