What do silver/gold tools do and how can I get them?

Silver and Gold Tools are upgraded versions of the regular versions of the tools, which usually confer a benefit like making it easier to catch things, better durability, etc. The acquisition methods vary depending on the tool.

Axes

Silver Axe

Upgrade: Always creates rare stump designs, lasts longer than the regular axe
Cost: 8 Medals
Acquisition: Buy at Tortimer's Island when it comes into rotation

Gold Axe

Upgrade: Chops down trees faster and is unbreakable
Cost: -
Acquisition: Buy 50 tree saplings at the Garden Center and Leif will give it to you

Fishing Rods

Silver Fishing Rod

Upgrade: Fish bite down longer on the hook, giving you more time between when the bobber sinks and when you can hit A to set the hook
Cost: 500 bells
Acquisition: Donate 30 Fish to the Museum and it'll come into rotation at the Museum Gift Shop

Gold Fishing Rod

Upgrade: Fish bite down even longer
Cost: -
Acquisition: Catch all 72 fish and then talk to Chip during the Fishing Tourney

Nets

Silver Net

Upgrade: Net radius is increased, effectively making it easier to catch bugs
Cost: 500 bells
Acquisition: Donate 30 bugs to the Museum and it'll come into rotation in the Museum Gift Shop

Gold Net

Upgrade: Net radius increased further
Cost: -
Acquisition: Catch all 72 bugs and then talk to Nat during a Bug-Off Tournament

Shovels

Silver Shovel

Upgrade: When hitting the daily money rock, it'll have a chance to produce gems instead
Cost: 500 bells
Acquisition: Donate 15 fossils to the Museum and it'll come into rotation in the Museum Gift Shop

Gold Shovel

Upgrade: You can plant Bells to possibly produce Money Trees
Cost: -
Acquisition: Buy 50 fertilizer at the Garden Center and Leif will give it to you

Slingshots

Silver Slingshot

Upgrade: Shoots 2 pellets at once
Cost: -
Acquisition: A present with 2 silver balloons will fly by with the Silver Slingshot inside

Gold Slingshot

Upgrade: Shoots 3 pellets at once
Cost: -
Acquisition: A present with 3 gold balloons will fly by with the Gold Slingshot

There is conflicting information on whether or not you need to have popped 16 regular balloons before the Silver/Gold Balloons will appear. Bidoof Crossing does not mention the requirement at all, the AC Wikia and AC Wikispace say it's only required for Silver and not Gold, and IGN says it's only required for the Gold Slingshot. I've personally not kept track of the number of balloons I've popped so far, so I cannot test myself.

Watering Cans

Silver Watering Can

Upgrade: Waters 5 spots at once (up, down, left, and right in addition to target)
Cost: -
Acquisition: Buy 50 flowers at the Gardening Center and Leif will give it to you

Gold Watering Can

Upgrade: Waters 9 spots at once (completely surrounding the target in addition to target itself)
Cost: -
Acquisition: Maintain a Perfect Town status for 15 days

Silver Tools

Silver watering can: Purchase 50 bags of flower seeds from Leif's gardening store. Can water up to five flowers at a time.
Silver axe: Rarely appears in island gift shop. Costs 8 medals. Always produces rare tree stump designs, and is more durable than a regular axe.
Silver slingshot: After shooting down at least 16 balloons, appears as a present under two silver balloons. Shoots two pellets at a time.

These silver tools can only be purchased from the museum after building the Museum Renovation public work, and cost 500 bells.

Silver shovel: Donate 15 fossils to the museum. Can produce gems from the daily money rock.
Silver net: Donate 30 bugs to the museum. Increased net size.
Silver fishing rod: Donate 30 fish to the museum. Fish bite longer.

Golden Tools

Gold shovel: Buy 50 fertilizers. Can produce Bell Trees.
Gold net: Catch all the insects in game and talk to bug catching tournament host. Greatly increased net size.
Gold fishing rod: Catch all fish in game and talk to fishing tournament host. Fish bite much longer.
Gold watering can: Keep your town in the "perfect town" status for 15 days, and you'll get the golden watering can from Isabelle. Waters 9 flowers at a time.
Gold Axe: Buy 50 tree saplings from the gardening shop. Never breaks, cuts trees faster, and sometimes creates a rare stump pattern
Gold slingshot: Found randomly in the sky, has three gold balloons. Shoot all down and inside will be the gold slingshot. Shoots three pellets at a time.

The Timer, Megaphone, and Toy Hammer do not come in any variant besides standard.