Pole recipes are generally unlocked by reaching certain market levels, at which point they can be purchased.

String-on-a-Stick
Owned at start.
Twigs x1, Rope x1, Metal x1

Real Fishing Pole
Driftwood x1, Plastic x1, Metal x1

Junk Rod
Twigs x1, Old Coin x1, Metal x3

Fisherman's Rod
Old Coin x5, Gold Flakes x1, Metal x20

Restless Rod
Twigs x10, Rope x5, Metal x50

Fancy Stick
Driftwood x10, Pearl x1, Metal x100

Sam's Rod
Obtained by fishing in Sam's Reservoir.
Purple Gem x1, Sky Crystal x10, Metal x10.0k

Total Materials Needed
Twigs x11, Rope x5, Driftwood x11, Old Coin x6, Gold Flakes x1, Pearl x1, Purple Gem x1, Sky Crystal x10, Metal x10174, Plastic x1

Tackle recipes are generally fished up in different areas.

Bent Nail
Owned at start.
Plastic x1, Plastic x1, Metal x1

Fisherman's Jig
Obtained by fishing in Backyard Pond.
Old Coin x1, Plastic x3, Metal x3

Double Plug
Obtained by fishing in Forest River.
Gold Flakes x1, Red Gem x1, Plastic x30

Speedy Spinner
Obtained by fishing in City Bay.
Rope x1, Green Gem x1, Plastic x50

Auto-Fly
Obtained by fishing in the Ocean.
Pearl x1, Yellow Gem x1, Plastic x50

Sam's Hook
Obtained by fishing in Sam's Reservoir.
Orange Gem x1, Plastic x20, Metal x50

Sky Fly
Obtained by fishing in the Sky.
Purple Gem x1, Sky Crystal x10, Metal x10.0k

Total Materials Needed
Old Coin x1, Gold Flakes x1, Red Gem x1, Rope x1, Green Gem x1, Pearl x1, Yellow Gem x1, Orange Gem x1, Purple Gem x1, Sky Crystal x10, Plastic x153, Metal x10053

Upon unlocking the Forge, all trade options are available. With each mod received, the gold cost increases, but the material costs do not change. There is no cap on how many times you can trade in each type of material, aside from any caps on the total number of mods you can get.

• Twigs x5000
  
• Old Coin x500
  
• Gold Flakes x200
  
• Rope x200
  
• Driftwood x200
  
• Pearl x100
  
• Red Gem x30
  
• Green Gem x20
  
• Yellow Gem x10
  
• Purple Gem x1
  
• Droplet x100
  
• Sky Crystal x10

In addition, there are also two special mods, which cost no gold. Dam Fishing is unlocked with Orange Gem x1, and divides the cast time in Sam's Reservoir by 10. Sky Fishing is unlocked with Purple Gem x1, dividing the cast time in the Sky by 10. You can get one of each gem with the first defeat of the area's corresponding boss, and more can be fished up in the respective areas.

To obtain all poles, tackles, and special mods, you'll need in total:

• Twigs x11
  
• Old Coin x7
  
• Gold Flakes x2
  
• Rope x6
  
• Driftwood x11
  
• Pearl x2
  
• Sky Crystal x20
  
• Plastic x154
  
• Metal x20227
  
• Red Gem x1
  
• Green Gem x1
  
• Yellow Gem x1
  
• Orange Gem x2
  
• Purple Gem x3

All excess materials can be freely used for mods or sold as needed. If I've missed anything, please let me know. Hope this helps!