Is there a systematic approach to find equilibria in sequential games? I know that one can use backward induction to find one particular subgame perfect NE. And I know that wherever possible one can represent the game in normalform and then find all NE.

But is there a systematic way of finding all extensive form specific subtypes of NE, like sequential or weak perfect Bayesian equilibria? What about the rest of the subgame perfect equilibria that can't be found by backward induction?

Do I always have to guess and proof that it really is the kind of equilibrium I think it is?
You can first find all NE. Then you check which ones are subgame-perfect. Then you proceed and check for which of the NE you can find beliefs that are consistent with the definition of PBE. You can go on and refine the set of equilibria further by kicking out all equilibria that do not satisfy the additional requirements of your stricter equilibrium concept.

If by "backward induction" you mean solving the game backwards subgame-by-subgame, then by definition you find all the SPNE (= all NE in which the equilibrium strategy profile also constitutes NE in all subgames).