A Trip to the Power Plant

Yeah, screw that long swim to Cinnabar, let's go to the Power Plant first.  Fly
on over to the Rock Tunnel Pokemon Center and go north from there.  Past a big
patch of tall grass, you should see a small beachfront where you can Surf.  Do
so, and swim right, then south when possible until you go past a small bit of
mountains blocking the land route, and you will reach the Power Plant.  There's
a single trainer here, with a Rhyhorn and Lickitung, both severely under
leveled compared to what you've faced before.  Once you've dealt with him, go
into the Power Plant.

Be on the lookout for Magnemite and Magneton, both of which occupy the Power
Plant and are excellent catches.  You can also encounter Pikachu and Voltorb
here, the former of which you should already have.  There are Electrodes
disguised as items, so be wary.  Go up and snag the first item ball you see for
a Max Potion, then continue on till you reach an intersection.  Go right and
follow the path there till you find TM17 - Protect.  After that, go up and left
along the north wall to get an Elixir.  Return to the intersection and go down.
At the next intersection, go down only if you want to fight and capture an
Electrode disguised as an item.  Otherwise, go right.  Head down at the next
intersection for a TM25 - Thunder.

Whew.  At the third intersection, go up to continue on.  Continue up at the
next crossroad and snag the Thunderstone on the right, but ignore the item on
the left, as it's an Electrode.  After that, just return and follow the path
till you encounter Zapdos.  Save your game before speaking to it.

Zapdos is a bird of a different nature.  It's both Electric and Flying, so it's
resistant Electric and immune to Ground, making it one tough bastard.  Your
best bet is Rock-type moves, its one and only vulnerability.  A good Rock
or Ground Pokemon (the latter having Rock moves) will stand a good chance
against Zapdos due to their invulnerability to this thing's powerful electric
attacks.  Zapdos' moveset consists of Thunder Wave, Agility, Detect, and Drill
Peck.

Be prepared to throw many, many Ultra Balls at this thing.  The Master Ball
should not, I repeat, should not be used here.  There are much more difficult
Pokemon to catch.  The best way to ensure capture is to either freeze,
paralyze, or put this bird to sleep, then start tossing your Ultra Balls.
Otherwise, you'll have a much more difficult time of things.

Finally, if you're having trouble, you can always leave and come back.  Zapdos
will still be here, waiting for you.  However, if you manage to catch it,
you'll have the first of the three legendary bird Pokemon on your side, not to
mention one tough critter.