I'm not sure what you're level is, but there are several ways to gain the offensive in a match as falcon. For me, if i would have known this i would have gotten to where i am much faster. I'll declare them in the order that i discovered them.
If you're both on the ground, (you should be dash dancing) dash forward and follow with a shffl'd stomp and try to land behind them. They can't do a noob shieldgrab, securing yourself and gaining distance. Not that effective, but should work out for you.
If you haven't noticed, falcon's dash attack isn't that great...... This compromises his amazing speed, or does it? Since you have so much speed, dash up and rather than dash attacking, grab him. It works wonders against people that hide in their shield, and you can combo the **** out of grabs.(I'll go more into grabs later)
Finally, i only recently discovered how amazing Nair can really be (its the flip kick). If you're playing against a non fast faller, you'll notice that many people like to advance by jumping to you. This is your chance to strike! Dash forwardand shffl that Nair! If you're having problems with that, just dash, short hop, hit the c-stick up, fast fall, L-cancel.(Work on it, its amazing)
Okay, i'll talk more about the grabs now. They're a great way to accumalate damage when you're opponent has low %. For example, if you're playing a marth and you have him in a grab, try throwing him down. You can either run up and grab him again, if he isnt going high enough, dashdance around until he rollows somewhere, and then grab him again! If you get good at this, its possible to chase him and lock him into grabs until he is ready to get a knee.
Okay, i forgot everything, if you have any questions, PM me