I use Mario as a secondary (I main Rosalina and struggle with the Falcon matchup, Mario has a better time with this), so I can provide some tips from Mario's point of view.
As
IcantWin
mentioned above, if Mario Dthrows you at 0%, you can just hold shield to shield his Utilt, as you land before it comes out. However, a good Mario knows this doesn't work at 0%, so he will do something else. A good Mario will do UThrow into Dair, which can't be escaped as Falcon because of his weight and fall speed. This can then be followed up with an Uair or 2 Bairs based on where you get launched. My advice for this to try and Smash DI Downwards to try and fall out of the Dair before it launches you. It is very hard to do, but it is indeed possible. If you cannot do this, do not spam air dodge, as Mario can still catch you with a Bair or Uair after your dodge finishes. The best thing to do here would be to jump and try and land safely.
Ok, so if Mario grabs you at 0%, does and Uthrow into Dair and nothing else, you have just enough percent to not be able to escape from his Dthrow Utilt combo. If Mario manages to land another grab, Dthrow Utilt will be coming. The best thing to do here is to DI upwards and away from Mario, or even Smash DI in the same direction. DO NOT AIR DODGE, as this gives Mario another opening to grab or smash you. Unfortunately, Falcon doesn't have a combo breaker move (like Mario's Nair) to escape combos like this, so things like Smash DI are vital.
In regards to preventing Mario from getting a grab, I am not a Falcon main, so maybe some of the Falcon gods on here can enlighten you more, but I find your Bair is decent at keeping Mario away. If you RAR (Reverse Aerial Rush) it, and Fast Fall in a Short Hop, you have no landing lag at all, and if Mario shields it, he can't punish it unless he perfect shields it, and even then he must be near perfect to get you. Don't just run or be aggressive with Dash Attack, Falcon Kick or Raptor Boost, as Mario can punish you with a grab if he shields them.
Overall, Mario is a bad matchup for Falcon, but it's not unwinnable. You a lot more speed than Mario. Use this to your advantage, and be deceptive. Dance Trotting can also be good for baiting grabs from Mario. If Mario is pelting you with Fireballs, try to Perfect Shield them. Get out there and show Mario the power of Falcon!
Here is a list of all the information you may need on everything I just said. I hope this helped!
DI and Smash DI -
https://www.youtube.com/watch?v=UIdyVK1aDtg
Dance Trotting -
https://www.youtube.com/watch?v=azRrqKUJ-5o
RAR -
http://www.ssbwiki.com/Reverse_aerial_rushwww.ssbwiki.com/Reverse_aerial_rush
Perfect Shield -
http://www.ssbwiki.com/Powershield