You are doing it wrong, and thinking it wrong. The problem with password hashing is not, and never has been, about collisions. The main problems with password hashing are speed (hash functions are too fast) and parallelism (a hash function is always the same as itself).
There is a lot of theory on password hashing. As an introduction, read this answer. The simple one-line summary is: use bcrypt.
Making anything better (i.e. more secure) than bcrypt is hard (some people are trying, and do it the right way: with an open competition, public research, and years of effort). However, making something which is much worse than bcrypt is very easy, and a lot of people achieve just that seemingly effortlessly. By your scheme with a combination of SHA-2 and SHA-3, you would be on the right track to add one more person on the depressingly long list of people who felt that creativity was a good thing in cryptography, and paid dearly for it.
(None of this says anything about SHA-2 or SHA-3 as generic hash functions. But, despite the names, hashing and password hashing are in fact very distinct activities, do not have the same requirements, and should not use the same algorithms.)