To unlock a character's talents, you will need to talk to specific NPCs as that character. For example, to unlock Lone's "Determined" talent, you will need to talk to Millie as Lone.

Some of them will charge money and some talents can be expensive. However, other talents are free!

Determined - Dash becomes a Weapon Attack. Go!
Talk to Millie in the Hideout after completing Terna Island Assault.

Blur - Dash Faster.
Buy from Callie in Dorfhole for 2000g.

Unstoppable - No more recoil when killing an enemy with Dash.
Talk to the drunk guy (Edward) in Halden after completing Pukamake Grotto.

Momentum - Never lose speed while Dashing.
Buy from Callie in Dorfhole for 4000g. Requires Blur.

Updraft - Tornado boosts you higher.
Buy from Prof. Gobel in Aexolix for 1500g.

Hurricane - Allows you to jump again while using Tornado.
Buy from Prof. Gobel in Aexolix for 5000g. Requires Updraft.

Mana Shield - Automatically reflect and destroy some bullets while focused.
Talk to Bello after buying all of Lydia's outfits.

Mana Jump - Double Jump has more height.
Talk to Samantha in Aexolix after completing Pukamake Grotto.

Stwongarm - Throw bombs farther.
Talk to Ogg'sean in the Hideout after completing Pukamake Grotto.

Gummi Bombs - Bombs bounce.
Buy from Megavitamin in the Hideout for 2000g.

Ethereal Bombs - Bombs pass through enemies.
Buy from Megavitamin in the Hideout for 3000g.

Impact - Big Mana Bombs have a blast radius.
Buy from Megavitamin in the Hideout for 800g.

Iron Will - Take half damage while focused.
Buy from Merlina in the Hideout for 4000g.

Flame Dragon Fury - Torpedo lasts longer.
Buy from Gustav in The Smiffy for 2500g.

Tidal Wave - Slide becomes a Breaker Weapon Attack.
Buy from Gustav in The Smiffy for 1200g.

Sea Dragon Sprint - Run faster.
Talk to Mo'i in Pukamake Grotto after returning to the Hideout.

Juggernaut - Breakable walls, floors, blocks, and pots break on touch.
Talk to the Orc Guard in Dead Man's Glade outside the Hideout.

Sky Dragon Grace - Improved steering for Torpedo. (Hold up/down)
Talk to Kimberly, the green fairy in Tricklethorn Vale.

Darkness - Shadow Attacks become Weapon Attacks.
Buy from the Lady Thief in the HIdeout for 3200g.

Train - Leap and Dark Step last longer if you hit an enemy.
Buy from the Lady Thief in the HIdeout for 1200g.

Void Walker - Dark Step lasts longer.
Talk to Ogg'sean (in the Hideout, not in the Necro Pits) after completing Vinegarden.

Dark Sprint - Dark Step ends with a burst of speed.
Talk to Lone in his room in the Hideout.

Slicer - Jump Slash lasts longer.
Talk to Anthony, the brown elf in the Skinless Man in Halden.

Reverse Cowgirl - Coup de Grace is no longer limited to once per jump.
Talk to Ku'i in the Hideout after completing Mudbottom.

Flip - Gives you a double jump. Do a flip!
Talk to Lydia in her room in the Hideout.

Firebombs - Bombs send out a wave of fireballs when they explode.
Buy from Megavitamin in the Hideout for 2400g.

Proximity Bombs - Bombs will explode when they're near enemies.
Buy from Gustav in The Smiffy for 2000g.

Dive Bomb - Dive Kick becomes a Breaker Weapon attack.
Buy from Ku'i in the Hideout for 2000g.

Boost Shoes - Run faster after using Dive Kick.
Talk to Kilzek in the Hideout.

Amplifier Circuit - Quadruples the power of reflected projectiles.
Talk to Abby in her room in the Hideout.

Blitz Thrusters - Adds forward thrust to your jetpack.
Buy from Anne for 2000g.

Fully Automatic - Shoot continuously instead of building focus.
Buy from Anne for 2000g.

Wave Mod - Splits charged shots into a wave of three shots.
Buy from Jeasel for 2000g.

Punch Mod - Increases charged shot damage but reduces range.
Buy from Jeasel for 2000g.

Shard Mod - Charged shots explode into pieces.
Buy from Gustav in the Smiffy for 2000g.

Mortar Mod - Charged shots become powerful Mortars.
Buy from Gustav in the Smiffy for 2000g.

Equip the following rings to get the associated benefit.
Increased Jetpack time: Rising Strike Ring + Pit Leaper Ring
Shots riccochet: Elastic Ring + Wall Rocket Ring

All of Penelope's talents can either be learned from Deathlord Schadel for Necro Tokens or found randomly in a Rank S+ Boss Chest in the Necro Pits.

Toxic - Damage enemies in Darkmist form.
Succubus Wings - Going up and down costs no energy. (Infinite flight)
Fallen Angel - Hold [Jump] to fall slowly.
Virulent - Move faster.
Unholy Crescent - Shadow Sickles no longer break.
Revenge - Damage enemies that touch you.