[RPG] How to use Tools in 5th ed D&D

For crafting non-magical items.

From the Downtime Activities on page 187 of the PHB:

Crafting

You must be proficient with tools related to the object you are trying to create (typically artisan’s tools). [...] For example, someone proficient with smith’s tools needs a forge in order to craft a sword or suit of armor.

So RAW it would appear that crafting doesn't actually require any ability checks. You just need proficiency in artisan's tools to be able to craft non-magical items in the first place.

But the DM might call for other ability checks that require artisan's tools.

From page 154 of the PHB:

A tool helps you to do something you couldn't otherwise do, such as craft or repair an item [...] For example, the DM might ask you to make a Dexterity check to carve a fine detail with your woodcarver’s tools, or a Strength check to make something out of particularly hard wood.

There are great new rules for these — and all the tool proficiencies — in the Dungeon Master's Tools chapter of Xanathar's Guide to Everything.

For each tool, these things are given:

• The components that make up the supplies or set of tools (and the note that proficiency means you know how to use them).
• Skills for which the tool proficiency grants advantage in certain situations, and suggested extra benefits for characters proficient in both the tool and the relevant check.
• A "special use" — a concrete thing you can do with this tool. For example, Brewing Supplies let you purify water during a long rest.
• Sample DCs for other activities the tool lets you perform. The word "sample" strongly implies that this isn't meant to be an exhaustive list (and such a limitation seems out of the spirit of 5E).

The Alchemist's Supplies can be used to give additional information in some Arcana and Investigation checks. They can also be used in Alchemical Crafting to make "alchemical items", of which there is a short list: a dose of acid, alchemist’s fire, antitoxin, oil, perfume, or soap.

The sample DCs are given for activities ranging from "create a puff of thick smoke" or "identify a poison" (both 10) to "Neutralize acid" (20).

Tinker's Tools can let you determine the age and origin of an object with a History check, or find how an object was damaged through Investigation. You can also repair damaged objects, at a rate of 10 hit points per hour of work (provided you have the appropriate raw materials).

The things you list (turn rock into gold or build a bomb) seem out of the power range of the various options explicitly listed, but they seem reasonable to attempt — if you set the DC appropriately high and the right raw materials are available and a reasonable amount of time is taken. Turning rock into gold might consume quite a bit of other valuable ingredients — or it might just take time. Under "Downtime Revisited" in the same chapter, there's this section on "Work":

Resolution. To determine how much money a character earns, the character makes an ability check: Strength (Athletics), Dexterity (Acrobatics), Intelligence using a set of tools, Charisma (Performance), or Charisma using a musical instrument. Consult the Wages table to see how much money is generated according to the total of the check.

So, you can use either of these tools to make money. You could just describe the Intelligence (Alchemist's Supplies) check as literally making money.