You've liberated Saffron City from Team Rocket's mad scheme, and you could probably use a bit of a holiday. We hear the fishing is good down south of Lavender Town, so grab your Old Rod and get set to drown a few worms. You ought to pack the Poké Flute, too. You won't be able to avoid Pokémon duels completely on this trip, but things will be far less hectic. Route 13 is more like a maze than a highway, but it is a pleasant place to walk and enjoy the great outdoors.
You'll need to use your Poké Flute to wake the Snorlax blocking the path here. But stop at that first building and get TM 39 first. It contains Swift, a good attack that virtually always hits (even if they're using Dig or Fly). The Poké Flute has the power to wake a sleeping Pokémon, in or out of battle. Pipe a sweet tune to rouse this Snorlax from its rather inconvenient resting place. It won't be very happy about the wake-up call, but if it attacks, you can try to capture it. This is the only other Snorlax in the game, so if you didn't catch the one in Route 16, here's another opportunity. With the way clear, continue south to Route 13. After the Snorlax, get the Super Rod in the 2nd building.
Route #12 encounters
Pidgey
40%
40%
15%
Pidgeotto
N/A
N/A
10%
Venonat
20%
20%
N/A
Bellsprout
N/A
35%
30%
Weepinbell
N/A
5%
5%
Farfetch'd
N/A
N/A
5%
Oddish
35%
N/A
30%
Gloom
5%
N/A
5%
Slowpoke
N/A
N/A
90%
Slowbro
N/A
N/A
10%
Moving on
If you want that Farfetch'd (finally, a Farfetch'd that isn't named Dux, but it's only in Yellow), or the hidden Iron, you'll need to get a Cut-equipped Pokémon to hack your way through the tree. Fortunately, TM 16 (the Pay Day skill) and the hidden PP Up (to the left of Trainer A on Route #13) are easier to get. Save that PP Up for one of the power skills: Thunder, Blizzard, or Fire Blast. Yellow players can also get a chance at a Slowpoke here just by using Surf and walking the waves of Route #12.
Now that you have the Super Rod, you'll be able to fish for more Water-type Pokémon than you know what to do with. So you might wish to ignore the less interesting ones like Goldeen. Evolved form Seaking is fairly tough, but with nothing but a handful of ordinary Normal attacks, she's not that intriguing.
Krabby is a fairly good Water Pokémon, and you can have fun beating things instantly (30% of the time) with his Guillotine technique. But Krabby's real selling point is that he can learn Cut, Surf, and Strength HM's, allowing you to have them all without using more than one slot on your bench.
#116 Horsea
#72 Tentacool
Horsea's evolved form Seadra is a solid fighter, and learns a good variety of Water-type attacks. But you probably won't want more than one or two Water-types on your bench, and there's a variety of better options. Wartortle, Cloyster, and Articuno top the list.
Tentacool mixes Water with Poison types, and learns a variety of good attacks in both (and has a high enough special to make good use of them). Built-in poison attack Acid does damage while lowering defense, which is nice, but Poison types still aren't stellar.
Venonat
Farfetch'd
Pidgeotto
Bellsprout
Gloom
Slowbro
Goldeen
Seadra
Krabby
