This randomized phase III trial is studying giving fluorouracil together with epirubicin and cyclophosphamide followed by paclitaxel and trastuzumab to see how well it works compared with giving paclitaxel together with trastuzumab followed by fluorouracil, epirubicin, cyclophosphamide, and trastuzumab in treating women with palpable breast cancer that can be removed by surgery. Drugs used in chemotherapy work in different ways to stop the growth of tumor cells, either by killing the cells or by stopping them from dividing. Monoclonal antibodies, such as trastuzumab, can block tumor growth in different ways. Some block the ability of tumor cells to grow and spread. Others find tumor cells and help kill them or carry tumor-killing substances to them. It is not yet known whether it is more effective to give combination chemotherapy before or after treatment with paclitaxel plus trastuzumab.