India accepts Shabbat and all other new days before Israel; Chicago accepts the days after Israel. At what point do we draw the line and say that to the east is Wednesday (in your case) and to the west is Thursday? There are different opinions regarding the halachic International Date Line.‎{&&1}‎ All agree that in all of Australia, days begin earlier than in Israel and much earlier than in Los Angeles.
You actually have a Wednesday: Tuesday night, when you leave, is halachically yom revi’ee,‎{&&2}‎ and you should count before, or soon after, takeoff. As you approach Sydney, you will lose much of Wednesday day and all of Wednesday night. Thus you will be unable to count omer with a beracha that day. You should indeed count Thursday morning (without a beracha) as you will have missed the night before.‎{&&3}‎ On subsequent nights, you may count with a beracha.‎{&&4}‎ If you arrive over the Australian continent before alot hashachar‎{&&5}‎ of Thursday morning local time, count then with a beracha.
There is a minority opinion that the halachic International Date Line is 180( longitude from Jerusalem. If you cross that line before alot hashachar of Thursday morning, count without a beracha at that time (in order to do the mitzva on the night of yom chamishi‎{&&6}‎ according to those opinions) and then again, in Australia, when you arrive (according to the opinions that it had still been yom revi’ee over the Western Pacific).
There are poskim who discuss losing a full day and what impact this would have on sefira and Shavuot.‎{&&7}‎ Although the International Date Line could cause a day to be extremely short, the possibility of losing a full day is statistically insignificant. In any case, it does not seem logical that one needs 49 sunsets to have 49 days of sefira. Since halacha has no choice but to set a halachic International Date Line, there is no reason one cannot jump into a new day without a new sunset upon crossing the line.