R1-12: Local Leve, Bone Ring (2 x Bone Chip), Bone Harpoon Butt (1 x Bone Chip), Copper Nuggets (4 Copper Ore or 4 Yellow Copper Ore), Brass Nuggets (3 Copper Ore 1 Zincs)
R12-15: Copper Earrings (2 x Copper Nugget), Silver Nugget (4 Silver Ore or 5 Silver Sand), Tier 1 Gem, Marbled Eye.
R15-18: Copper Ring (2 x Copper Nugget)
R18-25: Silver Francisca Head, Brass Earrings or Brass Rings. Start Brass Ring at r20 for ease of synth.
R25-28: Brass Rings or Silver Earrings, Tier 2 Gem, Electrum Nugget
R28-35: Silver Ring/Silver Needle, Mythril Nugget (4 Mythril Ore)
R35-39: Silver Chakram, Tier 1 Gemmed Ring, Silver Needle/Silver Ring, White Coral Arrowhead, Yarzonshell Harpoon Butt, anything you can get your hands on with decent SP
R39-44: Electrum Ring, Blue Coral Arrowhead, Tier 3 Gem
R44-50: Silver Francisca, Tier 2 Gemmed Ring, Red Coral Arrowhead, Tier 3 Gem.
Bolded means primary grinding synth.
Remember to save all your Silver Ingot you can get from local leve reward, you will need a massive amount as a reliable synth from R35-39 before you can do Electrum Ring reliably without hurting your pocket too much.
Silver Chakram is only few rank lower than Electrum Ring, why Electrum Ring from R39-44? Simple, Silver Chakram cost 1 Silver Ingot per synth, as cheap as this is you will be wasting more time gathering these outside of local leve reward you’ve been getting up till now. Electrum Nuggets might be pricey, but they come in bulk and abundant plust resell value far higher than Silver Chakram. Unless you also mine your own Gold Ore/Sand, don’t start with Electrum too early because they are pricey.
Try to make material (Nugget) while you still can get SP from it and save them for later. For instance, making Copper as soon as r1, Silver at r16 and Electrum at r26. Mythril currently has no use (only choker and wristlet) and are quite rare (Rarer than Gold) but you can attempt them at r35.
Lower level synth might be a bit off, lazy to check.
Training books that are relevant to Goldsmith synth:
Smelting, Gemcutting, Inlaying, Bonecarving, Fletchery, Nailcasting (very minimal involvement) -> can be bought from Goldsmith Guild
Bowyering (Armourclad Cavalry Bow) -> Carpenter/Tanner/Blacksmith Guild
Tacklecrafting (very minimal involvement) -> Carpenter/Culinarian Guild
Painting (Serpent Shield) -> Carpenter Guild
Leatherworking/Tanner -> Tortoiseshell (Optimal 25, Able to synth is 10 levels lower i.e. 15), Iron Preserve (Optimal 35, Able to synth 25)
Blacksmith -> Thunderstorm Axe (Optimal 35, Able to synth 25)
The 1.15a patch has changed SP gain substantially. It is a good idea to start doing your primary synth 2 ranks earlier than the guide if you can endure the high failing rate.
Silver Francisca is r48 while Electrum Ring is r44, from r44-48 Silver Francisca give much more SP per synth due to the patch 1.15a change, plus lower material cost, so it’s more efficient + effective grinding on Silver Francisca after r44.