What are the qualities of a good block cipher? Beginner, Intermediate, Advanced, Expert
I don't think we can really separate things into skill levels as such, but there plenty of targets to aim for in terms of security goals. You'll want resistance to:


Differential cryptanalysis


Truncated differential cryptanalysis
Integral cryptanalysis
Impossible differential cryptanalysis
Higher order differential cryptnalysis
Boomerang attacks

Linear cryptanalysis


Differential-Linear attack

Rotational cryptanalysis
Slide attacks
Meet in the middle attacks


Biclique_attacks

Brute force search
Side Channel attacks


Timing attacks

Algebraic attacks
Partitioning cryptanalysis
Attacks that don't exist yet (at least, publicly)


Some resistance can be achieved via:


Large enough keysize


Mandatory for simple brute force search

Large enough blocksize


A large enough size can preclude the conditions for a slide attack to be applicable

Avoiding using secret information in ways that leak timing information


No lookup tables
No shuffling of arrays according to secret information
No secret dependent rotations

High levels of diffusion that cause the avalanche effect quickly


Some other desirable traits:


Software efficiency on a wide variety of platforms (i.e. 8 bit, 32 bit, 64 bit)
Key agility


Ideally the key schedule is invertible
Ideally the actual keys used each round are extracted from the invertible key schedule in a weakly one way manner (first preimage resistant)



Don't forget to obey Kerckhoffs Principle

After you've done all that, you might like to make your construction tweakable

Some of the mathematical concepts related to cipher design are:


Information theory


Information Entropy

Finite Fields
MDS matrix


Matrix
Coding theory

Boolean functions


S-box (meaning NxM mapping, not a lookup table)
Bent functions