What is the thing Sakura is carrying?
As the comments by 曾気青昭雄 and Raxon suggest, this is indeed a bag. This bag was featured as the point in the Kero-chan check at the end of episode 10 of the anime:



In this segment, it is explained by  that Sakura uses this bag as a case to carry the baton she uses for her cheerleading. The letters "SAKURA" are embroidered on the bag. It was made by Sakura's father.