Lickitung can be obtained by trading a Golduck (FireRed) or a Slowbro (LeafGreen) to a man in the gate in Route 18. Lickitung will be the same level as the Golduck/Slowbro you traded over.
NO. STOP. WHAT ARE YOU DOING. GO AWAY. YOU DO NOT WANT TO USE THIS. GO GET A SNORLAX. OR A CLEFABLE. OR LITERALLY ANY OTHER NORMAL-TYPE AS LONG AS IT'S NOT. THIS. THING. STOP RIGHT NOW.
...Fine. It looks like you're dead-set on using this piece of crap, so we may as well make the best of it. Lickitung is basically the worst possible fully-evolved Pokémon you can get in the entire game excepting Farfetch'd and possibly Porygon, and even Porygon has potential after getting the National Dex as a Porygon2. What does Lickitung have? Absolutely nothing. You need to trade to even get it in the first place, for a Golduck in FireRed and a Slowbro in LeafGreen, and both are VASTLY more useful than this thing. Helpful tip: if someone is saying that Golduck is better than you, it's time to go home. Lickitung has no features that would make you use it over any other Normal-type. Pathetic attacking stats, decent at best defenses, and a Speed stat that ties it with Magcargo. MAGCARGO. Don't use this thing.
Gym #3 - Lt. Surge (Vermillion City, Electric-type): Why did you stall this battle so long? Anyway Lickitung is actually good here, since it will be at least level 33 and Surge tops at 23. Still, Raichu can do some real damage with Shock Wave.
Gym #4 - Erika (Celadon City, Grass-type): Stop stalling important battles. Unless you picked up an Ice Beam TM, don't use Lickitung here. If you did, it can still probably only kill Tangela.
Gym #5 - Koga (Fuchsia City, Poison-type): No no no no no. Lickitung would have to know Psychic, and you only get ONE TM of that, and you are way better off teaching it to something else, and also Lickitung still probably couldn't kill the Koffings. Weezing and Muk are both foregone conclusions: you're not getting through them.
Fighting Dojo (Saffron City, Fighting-type): HA no.
Rival (Silph Co.): Alakazam can probably be killed with Strength, but Lickitung has no chance against anything else, even if you give it Thunderbolt for Gyarados.
Giovanni (Silph Co., Ground-type): Rhyhorn, again, dies to Surf, but everything else will gleefully shit all over Lickitung's derpy face. Especially the Nidos, since they know Double Kick.
Gym #6 - Sabrina (Saffron City, Psychic-type): Again, nope. Lickitung is nothing but set-up fodder for her team, and you really don't want to say that you were swept by a Mr. Mime because of Lickitung, do you? However, Venomoth could maybe be killed with enough Flamethrowers.
Gym #7 - Blaine (Cinnabar Island, Fire-type): Fire Blast or Overheat from anything = dead Lickitung. Maybe beat Growlithe and Ponyta since they're weaker and Lickitung's bulk is not bad for them, but that's it.
Gym #8 - Giovanni (Viridian City, Ground-type): Surf can maybe kill the Rhyhorns, but everything else is carrying Earthquake by this point and even an Earthquake from Dugtrio will easily murder Lickitung. Stay away.
Rival (Route 22, pre-Elite Four): Pidgeot will kill you, Alakazam will kill you, the starter will kill you, Gyarados will kill you, Exeggcute and Growlithe probably won't kill you but are annoying and not worth it, and Rhyhorn should only be attempted if you have Surf.
Elite Four Lorelei (Indigo Plateau, Ice-type): No. Maybe Cloyster can be killed by Thunderbolt. But otherwise, no. Why is Lickitung in your Elite Four team again?
Elite Four Bruno (Indigo Plateau, Fighting-type): Both of the Onix die to Surf. Do not send out Lickitung against ANYTHING ELSE.
Elite Four Agatha (Indigo Plateau, Poison-type): The Gengars and Haunter are immune to basically everything Lickitung can do unless you have Shadow Ball, Arbok can tank through Lickitung's hits (and when the words "Arbok can tank" appear, you know something is horribly wrong), and Golbat can maybe be killed by Ice Beam but laughs at anything else Lickitung tries.
Elite Four Lance (Indigo Plateau, Dragon-type): Nope. Even the Dragonairs will shrug off your hits, even if you have Ice Beam. Don't.
Champion Rival (Indigo Plateau): Lickitung has no place in this battle unless you're severely overleveled. Pidgeot, Alakazam, starter, Gyarados, Arcanine, and Exeggutor all kill you before you can kill them, and even Rhydon is strong enough to tank a Surf and kill you.
Post-Game: HEY YOU HAVE THE NATIONAL DEX YOU CAN FINALLY EVOLVE LICKITUNG but wait. Nope. Lickilicky was introduced in Generation IV. Nevermind.
Lickitung has nothing in the way of level up moves. It learns no moves above 80 power, and that is Slam, which is terrible. It gets... Disable and Stomp, I guess? Its TM moves are much more interesting. Lickitung learns basically all the special attacking TMs. It doesn't have the stats to use them, of course, but let it have this. Flamethrower, Ice Beam, Surf, and Thunderbolt can all argue for a moveslot, as can Brick Break, Dig, and Shadow Ball. In the way of Normal moves, it gets Mega Punch and Mega Kick, Strength, Return, Double Edge, and...
...That's... that's it, actually. Wow. Um. So yeah. There's Lickitung. If anything, he's good as an HM slave, learning Cut, Surf, Strength, and Rock Smash.
Minimum stats are calculated with 0 EVs, IVs of 0, and a hindering nature, if applicable.
Maximum stats are calculated with 252 EVs, IVs of 31, and a helpful nature, if applicable.
What Nature do I want? Lickitung sucks in all areas, so Natures don't really matter. Neutral or Speed-lowering are probably best, though.
How good is Lickitung in a Nuzlocke? Horrible. Keep your Golduck or Slowbro and get an actual Route 18 catch. Every single thing Lickitung can do, something else can do better. Specially defensive wall? Chansey says hi. Hard hitter with good defenses? Kangaskhan and Snorlax exist, and Snorlax is basically a guaranteed catch. Special attacker with a wide movelist? Clefable learns everything Lickitung does and more, and has a Special Attack stat, too.
Neutralities: Bug, Dark, Dragon, Electric, Fire, Flying, Ghost, Ground, Ice, Normal, Poison, Psychic, Rock, Steel, Water