I think there is less of unifying theory than an assortment of artful tricks - and it's a pretty small assortment, too.

If you want to be able to reproduce every clever mathematical trick of the last 200 years you probably can (I believe in you!). But everyone short of Ramanujan learns them by studying many problems, repeated exposure, and either systematically or organically memorizing them. Afterwards, using them in a problem is more about pattern recognition. This is my experience as a physics phd-master-out ymmv