Finding the regular expression equivalent for the given finite automaton
Your DFA accepts the language of all words over $\{a,b\}$ in which the number of $b$s is of the form $3n+1$ (i.e., it is equivalent to 1 modulo 3). More generally, upon reading a word $w$, the DFA will be in state $q_1$ if it read a number of $b$s which is a multiple of 3; in state $q_3$ if it read a number of $b$s which is equivalent to 1 modulo 3; and in state $q_2$ if it read a number of $b$s which is equivalent to 2 modulo 3.

Using this description, it is easy to come up with a regular expression equivalent to your DFA. Rather than do that, let me give regular expressions for the language $L_{even}$ of words over $\{a,b\}$ containing an even number of $a$s, and for the language $L_{odd}$ of words over $\{a,b\}$ containing an odd number of $a$s:


$L_{even}$: $b^*(ab^*ab^*)^*$.
$L_{odd}$: $b^*ab^*(ab^*ab^*)^*$.


These examples should enable you to solve your specific problem.