Algorithm to find sequence of minimum moves to sort 13 card hand
This problem is harder than you might think.  Your operation of removing a block of adjacent cards and re-inserting the block somewhere else is known as a transposition.  Sorting by transpositions has been studied in the context of bioinformatics and is known to be hard.  Of course, if you limit the length to 13 cards, then efficiency becomes much less of a concern--even an exponential algorithm is likely to be fast enough.

I recommend breaking the problem into two related parts.  The first, lower level, is to write an algorithm for sorting by transposition on an array of integers, more specifically, on an array that is a permutation of the numbers 1..N.  This may be inefficient, but for small N, should be fast enough.

The second, higher level part, is to consider the possible arrangements of suits (clubs-hearts-spades-diamonds, ...).  There are at most 8 legal arrangements of suits, and fewer if one or more suits are missing.  For each arrangement of suits, it is easy to calculate, for each of the 13 cards, what position it should end up at, giving you an array of the numbers 1..13.  Then you can call the first part above as a subroutine.