This guide will be helpful whatever your budget may be.
Tools
Golden axe (with haste 3)
This is the starter tool, it shouldn't be used for long since upgrading to a better axe is easy and cheap. Also buying or making haste 3 potions, will in the long run cost you more than buying the next axe.
Jungle Axe
The Jungle Axe, unlocked at Jungle Wood 7 (craftable with 2 sticks and 3 Enchanted Jungle Wood) or bought from the auction house for no more than 10-100k should be the upgrade to the gold axe. It' s cheap price combined with the ability to break 10 woods in 1 hit should outclass the gold axe. Combined with a god pot it is a good starter tool and should help you advance through some Foraging levels until you unlock the next axe
Treecapitator
This is the last axe you can craft. The treecapitator is unlocked in Obsidian 8, allowing you to break up to 35 woods in a single hit. Combining to with a Haste potion allows you to break wood both with the ability and normally, making it the best axe out there. It costs 512 Enchanted Obsidian and a Jungle axe, totaling 1.3m coins. There is no upgrade to this axe so once you get it, it is the tool to get you to the higher Foraging levels.
Pets
Monkey
Although every other rarity than legendary doesn't really help your foraging xp gain, the legendary has the Evolved axes perk. At maximum level, it reduces the cooldown of the treecapitator by 50%, gaining you around 543,594xp per hour at lvl 80.
Ocelot
Considering that the Foraging wisdom boost(that gives you 30 foraging wisdom) is present at both epic and legendary rarity, epic rarity is better since it is easier to lvl up and gets you more xp. The ocelot pet, at vl 92 grants you only 442,107.6xp per hour, so the monkey pet is better.
Even though both the Giraffe and Lion pet are foraging pets, they don't help you lvl up and don't have any perk towards foraging.
Extras
Foraging EXP Boost
You can either get it by gifts or the god pot, level 1 grants you 5 Foraging wisdom, level 2 grants you 10 Foraging wisdom and level 3 grants you 20 Foraging wisdom.
Annihilation Cloak
Costs 480-500m and it grants you 2 Foraging Wisdom.
Destruction Cloak
Costs 100-120m coins and it grants you 1 Foraging Wisdom.
For your first setup: Get a gold axe with haste 3 and you will be find to grind for a little while before needing to upgrade.
For an early-game setup: You should definitely get a jungle axe and a Uncommon or Rare Ocelot ( if you can afford it get an armor that grands speed), that grants you only 25 Wisdom instead of 30.
For a mid-game setup: Get a Treecapitator with a epic Ocelot and young armor and you will be able to grind for a long time in order to get to the higher foraging levels.
For an end-game setup: The Toil Treecapitator with a Legendary Monkey and young armor is the setup you will need to get to the end of the foraging levels. Also you can get a Destruction cloak but the money to make it isn't really worth it for 1 Wisdom.
For an ABSOLUTELY maxed setup: This is absolutely not needed and no one with the right mind should spend this much for a foraging setup. Get a recombobulated Toil Treecapitator, with a LVL 100 Legendary Monkey pet, an Annihilation cloak, and a recombobulated young armor (because why not). This setup will cost you around 550-700m coins and shouldn't be bought for any reason since the buffs aren't really that big from the endgame setup.
Lastly, try to find a lobby where the Dark thicket is empty. Use young dragon armor run around oneshoting and breaking as many trees as possible. Haste really helps and should be used. I hope this guide is able to help you get higher in foraging and achieve your dream level!