there are a few ways they are :
Unfortinutlly you can not legitly catch Palkia in that particular game, you can migrate/trade it from pearl or get it on the GE.
in the ruins of Alph
you must get arceus at lv 80 + and go in rains of alph in the house.after you must select palkia dialga and geradina at lv.1
Yes
With a Poke Ball
in MT.Mortar
get an arceus and take it to ruins alph
its when you get a dialga palkia or giratina egg
Trade from Pearl or HeartGold/SoulSilver (Arceus event)
nothing, you just have three legendary Pokemon
You can't GET them, you have to trade them from pearl, diamond or platinum.
You can find Palkia in the spear pillar on the to of Mt.Coronet