1996 Mongoose size 19" Crossway 650 replacement size inner tube
According to the Bicycle Blue Book, this thing shipped with 700c wheels and 700x38 tires.

So, you need 700x38 tires (or something similar, e.g. 700x35) and a tube which is rated for 700x38 tires (these are sold by ranges, e.g. 700x38-45 tubes).