Some skills are obtainable as drops from Legendary Duelists from the gate or the ones that appear in Duel World, the skills that you can get are random from character exclusive skills, such as "Destiny Draw" of Yami Yugi and "Harpie's Hunting Ground" of Mai, to common skill drops, such as "LP Boost" skills and "Draw Sense" skills, that all characters can get.
You can fight any Legendary Duelists (LD) to get a chance to drop skills, the easiest ones being Tea Gardner, Weevil Underwood, Rex Raptor or Odion. Additionally it's better to duel against the lvl 40s since the chance of getting better cards, gems and skills are higher compared to the lower lvls. Rex is considered the easiest because he doesn't have any attack increasing cards other than his field spell, and most of his monsters have only about 1800 attack.
LDs you may avoid
Some Legendary Duelists like Kaiba, Joey, Bandit Keith or Ishizu are known to be difficult to farm, especially if you are using standard farming deck using cerberus.
Be careful on spending Gate Keys
Keep in mind that the colored gate keys, other than white, are used by 2 characters. If you plan on choosing one then make sure that you check the other character to see if you want to get a specific card from them, so that you don't end up using all of that colored key. → Details of Gate Key
Character featured events are chances!
There have been multiple event featured in the game, and during those event periods, players can challenge them without spending Gate Keys! The GameA website also shows decks to farm those LDs for a limited time with no set skill.
There's also a chance for a skill to drop when you play PvP. This only applies to Ranked Duels, and Casual Duels. Friend Duels won't give you any rewards. This can be a better option sometimes if you are low on keys, or don't want to spend any, and you can keep dueling other players without a limit. The common strategy that people like to do is to go to Casual Duels using decks that causes them to lose the duels fast, or simply keep passing the turns until your opponent beats you.
Decks to farm skills
The main thing that you'll want to achieve is a high score, and can farm fast and consistently. Cerberus decks are often used due to how easy it is to get attack more than 9999 damage which gives 3000 pts, and the cards needed in a Cerberus decks aren't difficult to obtain. → How to get a high score
High Score Scorpion
I strongly recommend you to use this deck for the following three points:
No card from booster packs needed.
Can get around 6,000 score against Rex stably.
No skill is required that means you can use any character with this deck.
Basically your Yami Yugi, Weevil Underwood, and a character who has already acquired drop "Draw Sense: Light" skill, can handle a deck with cerberus and put 19 spell cards which includes stop defense works. Their "Sorcery Conduit" skill and "Draw Sense: Low-Level" skill allows you to get the monster to your hand. This is known to be an easy way to achieve a high score. → High Score Cerberus: deck recipe
Tethys deck can be used to farm easily farmable LDs like Rex Raptor pretty fast and consistently, and it also doesn't require any character specific skills so you can use it on any characters. Unfortunately, Tethys was only available through an event, but if there are future obtainable ways and you get 1 or 2 copies of her, you can use her in the Tethys fairy deck for skill farming.
Last Gamble Can be used in your Main Phase, starting from the fifth turn. Upon activation, reduces your LP to 100 and discards 2 cards from your hand. You roll a die and draw that number of cards. This skill can only be used once per Duel.
Creator Can be used if your Life Points drop below 2000. In the Draw Phase, instead of doing a normal draw, a powerful card, from outside of your Deck, is added to your hand. This skill can only be used once per Duel.
Three-Star Demotion Can be used when your Life Points are 1,000 or below. Until the End Phase, reduce the Level of all monsters in your hand by 3. This skill can be used to "Suijin, "Kazejin", and "Sanga of the Thunder," regardless of your remaining Life Points. This skill can only be used once per turn.
Life Cost 0 You can use the skill when your Life Points are 1000 or less. You do not have to spend Life Points to activate a card until your opponent's next turn ends. You can only use the skill once per Duel.
Firm as the EARTH Can be used after your Life Points are at 3000 or below. Play the Field Spell "Gaia Power." If there already is Field Spell on your side of the Field, rerturn it to your hand. This skill can only be used once per Duel.