I have a couple of doubts regarding Libor curves from an academic perspective. I need to construct these 3 curves(1,3,6M LIBOR ) separately. For an academic exercise, I want to construct these 3 curves separately and then show them that the rate at any tenor extracted fromthe 1M curve and 3M curve are not the same. By using this, I want to show that there exist a basis between 1 & 3 , 3 & 6 etc, when the tenor point goes beyond a point in time. I need to do this exercise for just USD curve for now, but would like to extend it in the future once I understand the workings of this curve and bootstrapping. Please do help me or suggest me a suitable paper so that I can kick start my exercise.

A Note on Construction of Multiple Swap Curves with and without Collateral; Fujii, Shimada, Takahashi. Search SSRN for it. Also other papers by these three. I'm not a big fan of the way they set up their theoretical framework (in my opinion Cooking with Collateral by Piterbarg is much more sound; search Risk for that), but they go through the bootstrap in quite a lot of detail.I'm not quite sure what you mean by constructing the curves 'separately': for USD you would normally bootstrap the discount curve first (from OIS), then the 3m curve (from IRS), then curves for other tenors (from tenor basis swaps).