OK (DEEP BREATH) First,to get fly beat the gym leader in cianwood city
and talk to the lady outside the gym she'll give you fly.

Second to cross the river you must have all eight gym badges and have
waterfall and surf. you get surf in ecruteak city when you beat the dancing kimonos and talk to the guy and his ryhdon and he'll give you surf. to get waterfall , the first maze in the cavern to get to blackthorn city there should be two passage ways. one on ice and one on land. take the on on ice and you should find waterfall at the end.

now to get to the pokemon league you must surf east (right)of new bark
town keep going until you find land. there should be a cavern and a water fall. surf west shortly to find a moon stone and north to the water fall to get to the road leading to the pokemon league use waterfall by going up to the water fall and pressing a. your pokemon must know waterfall. surf andkeep going until you get out of the cavern.
now you are getting closer to victory road. keep going and battle some trainers.keep going until you find a somewhat large building. go in this is the enterance to victory road. go through victory road. and you should find the pkmn league. (HAND CRAMP)

i think u get fly in the house nearest to the fifth gym
which river are u talking about? (johto or kanto?)
for johto, i didn't cross any river in my sister's silver version for the tunnel to league.
for kanto, sorry i didn't get there yet