Pidgey can annoy you with Sand-Attack, which is certainly annoying. Try and switch the
affected Pokemon out when Falkner switches for his Pidgeotto, which is at level 13. Level
13 is actually pretty tough for this point in the game. They did a good job at making
Falkner a bit more of a challenge, since he used to use only a level 7 Pidgey and a level
9 Pidgeotto.
Pidgeotto likes to use Tackle, which isn't too strong normally, but Pidgeotto's rather
high level may make it a bit painful nonetheless. It also has Roost in its arsenal, which
heals its HP and restores it by 50% of its max. Don't underestimate that, because it can
use it whenever it likes to heal its HP back!
If you're having trouble beating Falkner, go catch a Mareep to the southwest of town on
Route 32, level it up a few levels, and then just send it out against Pidgeotto. Hopefully
Static will activate when Pidgeotto uses Tackle, paralyzing it, cutting its Speed by 75%
and hopefully preventing it from using Roost due to paralysis. Mareep doesn't even need to
know Thundershock (learned at level 10) to be useful.
After beating Falkner, he'll give you the Zephyr Badge, let you use Rock Smash outside of
battle, will let traded Pokemon up to level 20 obey you without question, and even give
you that annoying Roost move in the form of TM51. That's a nice move to teach your
Hoothoot or Pidgey if you plan on using it, though you may also wish to save it for other,
more useful Pokemon. (Hoothoot is a good option though, due to its already high defenses
and the fact that it can use Hypnosis.)