Edit: A gentle book for character theory is G. James and M. Liebeck's Representations and Characters of Groups. Its chapter 23 is entirely to devoted to all of these different ideas (as well as Frobenius's original motivation) without using anything fancy. It has many examples, many exercises, and solutions to many exercises. I think it is useful for both group theorists and people who just want to use representations.
For fancier things: I. Reiner's Maximal Orders has nice coverage of the division algebras associated to finite group algebras (and their maximal orders). My favorite textbook treatment of the numerical invariant of the division algebras are Albert's Modern Higher Algebra and Structure of Algebras, but some people think they are old fashioned. Recent treatments will often get lost in technicalities that are irrelevant to group algebras. You might be OK with one of large textbooks on algebra; many have a chapter on Central Simple Algebras and the Brauer Group. MathOverflow readers will probably like Rowen's Algebra: A Non-commutative View.