Lulav on First Day Sukkoth

Should we take the lulav when the first day of Sukkoth falls on Shabbat?

Yes, if you are a resident of Erets Yisrael. This is explained explicity in the Mishnah and both the Talmud Yerushalmi and Talmud Bavli. This was the custom for hundreds of years in Eretz Yisrael and is still practiced by a few. For more information, see The Misswa of Lulav on Shabbat.