Both Golduck and Slowbro can be found at Cape Brink on Two Island. However, their pre-evolutions (Psyduck and Slowpoke) can be found much sooner with the Super Rod.
Getting Your Lickitung
The Pokémon you need to trade to receive a Lickitung will differ between FireRed & LeafGreen, but neither game has Lickitung available in the wild.
Lickitung Trade Location: Route 18
Once you have the Golduck (FireRed) or the Slowbro (LeafGreen), you can head to Route 18 just left of Fuchsia City: Step 1: Head west of Fuchsia City and enter the yellow brick building. Step 2: Walk up the stairs at the bottom of the building. Step 3: Talk to the man at the top of the stairs and agree to the trade. Step 4: Select the Pokémon you’re trading and receive your Lickitung.
How To Get Golduck (FireRed)
Golduck can either be caught on Two Island, or you can evolve a Psyduck into Golduck. Psyduck appears much sooner in the game, once you get the Super Rod, and will evolve into Psyduck at level 33. But Cape Brink is the best place to find wild Golduck to catch. Cape Brink is just north of Two Island, and there is a patch of grass with a 5% chance of encountering a wild Golduck. If you’re super unlucky, there are also many high-level Psyducks in this grass that you could catch and train. More Golduck locations: Alternatively, Psyduck can be found in Cape Brink and a number of other post-game locations. However, if you can get to those locations, you can find a Golduck anyways. So here are a few early-game locations where you can find wild Psyducks as long as you can use the Super Rod.
How To Get Slowbro (LeafGreen)
If you have LeafGreen, you’ll need to trade a Slowbro to the man on Route 18. You can either catch a Slowbro on Cape Brink, the Seafoam Islands, or you can catch and evolve a Slowpoke. You’ll be able to catch a Slowpoke much earlier in the game, but if you’ve already defeated the Elite Four, then catching Slowbro is the better option. If you’ve already made it to Cinnabar Island then you can probably catch Slowbro. Slowbros can be found on the Seafoam Islands or at Cape Brink, north of Two Island. Best Slowbro locations: Alternatively, you can also find Slowpoke on Cape Brink. But if you’re all the way on Two Island you might as well just catch a Slowbro. So for an early-game option, here are the best locations for catching Slowpoke, as long as you have the Super Rod: