LOL. I'm not online for a few days, and suddenly threads are being tossed to people like free Tootsie-Roll Pops?

Anyways, Eggz should handle most of the stuff, just because he's Eggz. He has a combo vid and actual skill to his name; I just write lots of stuff. XD I'll help out when I have the time, though.
When it comes to jabs, Mario really, really needs it. Frankly, I'm convinced that Mario's jab game is probably just as critical as his grab game and his combo game--if anything, Mario's about combos, grabs, edgeguarding, and an awesome jab game.
You're supposed to be using jabs extensively, mostly because you should be within jab range anyways. We all know that Mario is a close-range fighter; his entire game revolves around getting those grabs -> combos, and jabs do a godly job of pressuring opponents and opening up those grab/combo opportunities.
The thing is, if you've ever seen Wes' Samus play before, or a high-rushdown/pressure Flaco player, you'll know that jabs are important. Both Wes and decent Flaco players love to pressure with jabs--they'll come in close (usually after some sort of projectile rushdown, but not always) and proceed to jab. Jab into smashes, jab into tilts, jab into spot-dodge into more jabs, jab into dash away/behind them + shffl'd aerials, jab into grabs. . .all sorts of good stuff.
Basically, using jabs can snuff out opponent's attacks, pressure them if they're shielding, and can lead to a whole bunch of nasty stuff (grabs, smashes, tilts, aerials, dashdancing, etc.) simply because the stun time is so good. In short, jabs are essential in leading to other attacks/grabs, and especially help in opening up combo opportunities.
Of course, Mario's projectile rushdown isn't that great, so you have to think of other ways to get in close to initiate your grab game. And that's a whole different topic. But again, jabs are critical for any Mario player, so be sure to incorporate them into your game.
If you're looking for specifics, I can think of:
- jab -> d-smash
- jab -> u-tilt
- jab -> grab
- jab -> jab -> d-smash or u-tilt or grab
- jab -> dash away, dash back -> shffl'd aerial/grab/smash
- jab -> dash through and behind them -> reverse f-smash/dash back + grab/ SHABAWD (this works especially if they're shielding)
That's basic stuff. I'm sure there's other stuff that I've forgotten too; Eggz and whoever else wants to hop in can suggest other things.

Just be sure not to get too carried away if the opponent's shielding; if you start using high-lag attacks then they can obviously take advantage of that when they're shielding and shieldgrab you.