Choosing your harp is half the battle.

If you want to play blues harp, you need to be able to "bend" notes to get that "bluesy" sound.  To play cross-harp you flatten notes in your scale by simultaneously choking off some of the air and sucking or drawing air through the reed chamber.   

Your first step is to get a good quality harmonica.  The reeds need to be air tight and perfectly aligned on the comb.  A quality harp will have proper reed set up out of the box and the reeds will respond well to both blowing and drawing notes. Getting a cheap harmonica and adjusting the reeds is not something an average person can or wants to do.  It requires finesse and experience and you will very likely ruin several reed plates before you get the hang of it.

Hohner makes some quality harmonicas, but my preference is the Tombo/ Lee Oskar.  I've been playing them for over 20 years.

Bend on!