The reason I won't create an own page for Enchanting, Alchemy and Taming is that these skills are extremely easy to level and require almost no active effort. These will probably be the first skills you max out, alongside Runecrafting (not covered as it's only cosmetic). Alchemy just requires money, Enchanting needs you to do daily minigames and Taming is entirely passive, but you can speed it up.
You get Alchemy experience every time you break a netherwart crop or collect a potion from a brewing stand. While Netherwart farming might help you leveling this skill, the fastest way is to brew expensive potions.
Most people use either Speed 6/7 potions (Enchanted Sugar Cane) or Weakness 6/7 potions (Enchanted Fermented Spider Eye), but Strength 6/7 potions (Enchanted Blaze Rod) are also an option.
What you will do is simply set up a lot of brewing stands, brew the potions and collect them. Reminder of how to brew the potions:
3 water bottles in each brewing stand
1 Netherwart (skip if you use Enchanted Fermented Spider Eye to brew Weakness potions)
Put in the ingredient mentioned above depending on what potion you want to brew
Put in 1 Glowstone Dust or Enchanted Glowstone Dust (I'll explain later)
To maximize the xp you get you should definetely be using a god potion and a booster cookie, as each boosts your skill exp gain by 20%.
It is advised to also wait until Derpy becomes mayor, because he will further boost the xp by 50%, making it faster and cheaper to level up the skill.
If you don't care about money, you should definetely be using Enchanted Blaze Rods, they are not the most cost effective but they do give the most xp out of all the potions.
Minimizing needed time
In addition to following the steps in the Maximizing Xp section, you can also choose Enchanted Fermented Spider Eyes over the other 2. That's because you can skip step 2 of brewing the potion, you don't need to put in Netherwarts.
You can also skip step 4, but that will greatly increase the cost.
Use all available ways to boost your xp that don't increase cost (no Enchanted Blaze Rods). Also make sure you always have an Alchemy pet equipped while collecting potions. There are 3 Alchemy pets in the game: Sheep pet, Parrot pet and Jellyfish pet.
Out of those 3 the Sheep pet has the highest demand, but it might nevertheless be worth it to level a Parrot or Jellyfish pet. Equip the pet with an all skill exp boost item and level it to level 92. At level 92 the pet has 50% of the xp needed to get it to max level, but most people think it's more. That means the price should be kinda close to that of a level 100 pet. After getting the pet to level 92 sell it (or wait until derpy is gone to let the prices recover) and get a new pet to level up.
The last thing you can do to decrease the cost is to increase the level of the potion by using Glowstone Dust or Enchanted Glowstone Dust. Doing that will increase the NPC sell price.
When Derpy is not Mayor you could also sell potions on the ah instead of selling to the NPC, for possibly triple the price, but keep in mind that this doesn't work if you do a bunch of potions at once because the demand is low.
With maximized cost effectiveness leveling your Alchemy skill from level 0 to 50 takes about 20m coins (with Derpy) or 30m coins (without Derpy). You will have to brew 568/852 potions (using Sugar Cane or Spider Eyes, with/without Derpy).
This one is a lot easier than Alchemy. All you have to do is get Enchanting Level X (10) by enchanting stuff. Once you've done that you gain access to the Experimentation table, which will make reaching Enchanting LX (60) easy. Place it down and click on it. You can do the Superpairs game by paying xp levels once a day, granting you loads of Enchanting xp and if you're lucky expensive t6 or t7 enchants. You can also spend some more xp levels and Bits to do it again, up to 3 times a day.
As you level up the skill you will not only unlock "harder" versions (giving better rewards) of the Superpairs game, but also 2 "Addons". Doing those will enable you to uncover more tiles in the Superpairs game.
To level up faster you could use an epic or legendary Guardian pet, but that will hardly be necessary considering how easy leveling up this skill is. Instead you could use the massive amount of enchanting xp to level up some of your other pets. To do that simply give them an all skill exp boost item and equip them before doing your daily experiments.
The most important thing to do in order to level this skill is to have a pet equipped at all times. Every time one of your pets gains pet xp, you gain taming xp as well. At the same time leveling your Taming skill will make pets gain more pet xp.
While you can level this skill completely passively, theres a few things you can do to make it go faster:
Always have a pet equipped that fits the type of xp you get (e.g. Combat pet for dungeons)
Always give your pets xp boost items that fit their type or an all skills xp boost
Always have a pet that's holding an exp share item, as that will let you have 2 pets gain pet xp at the same time (if you have multiple pets exp shares they won't gain more pet xp than with just one, but instead the pet xp will be split between the 2)