running shoe vs. training shoe

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Hi,

I am needing to buy sneakers again, and I started running, well Pooch to 5k training. I do lots of kickboxing,(turbofire and turbokick) dance workouts, Insanity - lots of plyometrics.

I am thinking of getting a running shoe, maybe another pair of shox, and a training shoe for my in house cardio.

Not sure what to do or get.

any ideas on what the differences are for a running shoe and a training shoe?