A lot of people complain about skills in 5e because the benefits of being trained in a skill are miniscule. A meager +2 bonus at first level that scales upward to +6 at the highest echelons of play? Even the most specialized characters can only push that to +12 with skill expertise.
Now, I'm of the mind that:
(a) the base math of proficiency is fine the way it is because you don't need to rely on larger numbers to demonstrate an increase capability with a skill;
(b) skill expertise is bad design because it butts against bounded accuracy and encourages the idea that the only real form of advancement is through numerical increases;
(c) there's a better way to handle skill checks.
Let's start with treating proficiency as something other than a +X to your d20 rolls.
Proficiency: Narrative, Not Numerical
Characters who are proficient in skills aren’t merely a few percentage points better than their non-proficient counterparts. They are leaps and bounds ahead of them, even if that isn’t reflected in the math. Proficiency represents a degree of training and specialization that non-proficient characters lack. They are always better than non-proficient characters regardless of the objective numbers involved. Success and failure are measured both quantitatively (exact numbers) and qualitatively (what happens as a result of the roll). Proficient characters can achieve higher successes and mitigate the consequences of failures. Two characters can be identical in terms of their dice rolls, but the proficient character possesses a level of excellence that non-proficient characters lack.
Benefits of proficiency include:
Automatic Successes. Proficient characters automatically succeed at some tasks without rolling the dice.
A blacksmith is forging a sword. He doesn't need to roll. The investment of time and raw crafting materials allows him to forge a sword.
Mitigated Consequences. Even on a failed roll, proficient characters experience reduced consequences. (You aren’t rolling if the result of the roll is “nothing happens,” right?)
A thief wishes to scale the side of a building and enter the lady’s bedroom to investigate her heirloom jewels. The GM rules that the character must make a check or risk awakening her ladyship. A non-proficient character fails the check, and the lady starts awake, calling for her guards! Should the non-proficient character fail, her ladyship rouses drowsily.
Style and Substance. Proficient characters demonstrate their skill when rolling regardless of outcome. While non-proficient characters may fumble through tasks, proficient characters handle them skillfully even when fortune turns against them.
A narrow beam spans a pit that an adventurer is determined to cross. He must make a check to cross, lest he fall into the pit. A non-proficient character succeeds at the check, tottering feebly across and nearly falling several times. A proficient character fails at the check. Though his stride is sure, he happens to look down and is struck by vertigo, and he loses his footing.
Expertise. Proficient characters have specialized training and knowledge, so they can attempt tasks that non-proficient characters cannot.
A scoundrel seeks illicit entrance into a manor. The door is locked. With the appropriate proficiency, he can pick the lock. Without, he’ll have to break down the door or find another manner of ingress.
Partial Successes. Even when failing, a proficient character is likely to achieve some of what he set out to do.
A wizard deciphers a faded text. As he is proficient, even if he fails, he’ll translate some of the text—enough to understand the basics and perhaps ask a clarifying question or two.
Questions and Answers. Proficient characters know things about their fields of knowledge. More often than not, they can ask questions and get truthful answers from the GM--no skill check required.
A kindly healer examines a sickly beggar covered in sores.
"Do I recognize this disease?" he asks.
"It's clearly the scabrous pox," the GM responds.
"What's the treatment?" he asks.
"You can purchase a simple balm. A few days of applying that regularly will cure what ails him." Again, the GM answers at once, no roll required.
When should a GM implement these benefits? That is a judgment call. In general, proficient characters should be able to show off their proficiency regularly.
Simple and Expert Tasks
Simple tasks are tasks that anyone can try. They are those standard things adventurers are good at because they encounter them in their travels. Examples of basic tasks:
Sneaking past a dozing sentry.
Sweet talking a merchant into showing you his “special” wares.
Searching a room for signs of traps or treasure.
Riding a horse, climbing a tree, swimming across a river.
Foraging and hunting.
Expert tasks require proficiency because they’re too complex or too specialized for just anyone to handle. Examples of expert tasks:
Picking a lock with thieves’ tools.
Deciphering arcane runes.
Identifying mushrooms as edible or poisonous.
Forging a suit of armor.
Playing a musical instrument.
Why Does This Work?
This is how I (Helpful NPC Thom) run my games, and they're better for it. It lets the characters feel like competent specialists without resorting to numerical inflation. The characters can feel like adventurers who are well-trained in their abilities without changing a single rule.