For the jab to grab, you press grab immediately after one of the jabs (almost as if you're pressing a and r at the same time, but r slightly after). So it would be something like a-a-a-a-a-a-a-a-ar.
For the up-B twice across the ground, lets say you were going to the right. What I do is up-B to the right first, then angle my second one downwards. Basically, while on the ground, an up-B downwards makes you go sideways in the direction your controller is more pointing in (so down right will make you go right, down left makes you go left). Straight down makes you go right for some reason. So doing two up-B's angled downwards and to the right that are over 45 degrees apart will result in two up-B's in one direction.
Edit for Rickety's question:
Z-cancelling was known well before melee. If I remember right, Nintendo had a site for various advanced techniques including Z-cancelling, so it was known as soon as the game was released.