1.楼主此题错矣.
would rather加宾语从句时才会用虚拟语气.
规律如下：
1）对过去的动作虚拟时,从句用过去完成时.
如：I would rather (that) you hadn't come yesterday.
2) 对现在或将来的动作虚拟时,从句用一般过去时.
如：I would rather (that) I were a millionaire.
I would rather (that) it rained tomorrow.
2.would rather还有如下短语：
1) would rather (not) do sth 宁愿(不)做某事
2) would rather do sth than do sth = would do sth rather than do sth
宁愿做某事也不愿做某事
3.所以该句应该是：
I would rather go to school today than go swimming tomorow.
即前后都用动词原形.