Go into the dojo and speak to Ginryu. He asks Ryoma to fight his students using only his sword, so he can have a look at Ryoma's level.
#1: Special Ability - Dragon Fang Flash, Beautiful Technique - Misty Sword, Special Ability - Wild Dance of the Divine Dragon, Beautiful Technique - Wind Circulation
The first part of this is the Sword Fighting Style special ability, which launches a high speed powerful attack.
Hold down R2 for a little while and then release to launch an attack. You can use the left analogue stick + square when your first attack hits to launch another attack on an enemy. You can continue doing this, but your heat gauge is drained for each additional attack you hit.
Then next part of this is the "Beautiful Technique - Misty Sword" ability for the Sword fighting style. Press triangle while evading (X) to use.
The next part of this is the Wild Dance Fighting Style special ability which launches gunshots and sword attacks wildly. Hold down R2 to use.
The final part of this is the "Beautiful Technique - Wind Circulation" special ability for the Wild Dance fighting style. Press L1 when an enemy attacks you to evade the attack.
After completing the tutorial, the seal on the "Beautiful Technique - Misty Sword" ability in the Sword sphere grid and the "Beautiful Technique - Wind Circulation" ability in the Wild Dance sphere grid are removed and you can now purchase them with one of your souls. You can also now use the Dragon Fang Flash and Wild Dance of the Divine Dragon special abilities in battles.
#2: Beautiful Technique - Flashing Sword, Beatiful Technique - Tsukimizake
When you have a 3 star sword equipped (you cannot do the next training with a sword that has a higher or lower star rating), talk to Ginryu. You get to choose which fighting style to use - Top option for Sword, bottom option for Wild Dance.
When you choose the Sword Fighting Style, the training is for the "Beautiful Technique - Flashing Sword" ability. While holding R1, press triangle just before an enemy hits you with an attack to perform a strong counter.
After completing the tutorial, the seal on the "Beautiful Technique - Flashing Sword" ability in the Sword sphere grid is removed and you can now purchase it with one of your souls.
Next, speak to Ginryu again and you'll begin the Wild Dance tutorial for the "Beautiful Technique - Tsukimizake" ability. Press square while evading to attack with the sword.
After completing the tutorial, the seal on the "Beautiful Technique - Tsukimizake" ability in the Wild Dance sphere grid is removed and you can now purchase it with one of your souls.
Speak to him again after this and he asks you to compete against his student. You have to get the battle grade "Expert", at least, to pass (you are graded after every battle in the game - it's that screen you get right after you win a fight). You can fight using the Sword (top option) or Wild Dance (bottom option) fighting styles.
#3: Beautiful Technique - Weather Vane, Beautiful Technique - Tengu Shadow
When you have a 4 star sword equipped (you cannot do the next training with a sword that has a higher or lower star rating), talk to Ginryu. You get to choose which fighting style to use - Top option for Sword, bottom option for Wild Dance.
For the Sword Fighting Style, the training is for the "Beautiful Technique - Weather Vane" ability. Press triangle when an enemy attacks you while you're defending (L1) to repel the enemy's attack (this only seems to work while an enemy is performing a combo attack).
After completing the tutorial, the seal on the "Beautiful Technique - Weather Vane" ability in the Sword sphere grid is removed and you can now purchase it with one of your souls.
For the Wild Dance Fighting Style, the training is for the "Beautiful Technique - Tengu Shadow" ability. While you are attacking, press L1 when an enemy attacks you to evade the attack.
After completing the tutorial, the seal on the "Beautiful Technique - Tengu Shadow" ability in the Wild Dance sphere grid is removed and you can now purchase it with one of your souls.
Speak to him again after this and he asks you to compete against 5 of his students. You have to get the battle grade "Master", at least, to pass. You can fight using the Sword (top option) or Wild Dance (bottom option) fighting styles.
#4: Bullet Dodge Mastery, Crowd Dispersion
After completing #3, leave the dojo and then come back inside again and talk to Ginryu. You need to have a 5 star sword equipped (you cannot do the next training with a sword that has a higher or lower star rating).
He asks you to compete against his student. You have to get the battle grade "Master", at least, to pass. You can fight using the Sword (top option) or Wild Dance (bottom option) fighting styles.
When you win the fight with a battle grade of "Master", at least, the seals on the "Bullet Dodge Mastery" ability in the Sword sphere grid and the "Crowd Dispersion" ability in the Wild Dance sphere grid are removed and you can purchase them with your souls.
#5: Beautiful Technique - Dry Leaf Disturbance, Heat Haze Circulation
After completing #4, leave the dojo and some students go out for a talk with Ryoma. After they finish, go back to the dojo and talk to Ginryu.
Select the top option to do the next training.
The first part of the training is for the "Dry Leaf Disturbance" technique. While holding R1, press circle when an enemy attacks you with a sword to defend against their attack.
After doing that ability 3 times, defeat all the opponents.
After completing the tutorial, the seal on the "Beautiful Technique - Dry Leaf Disturbance" ability in the Sword sphere grid is removed and you can now purchase it with one of your souls.
Speak to Ginryu again to begin the next part of the training. He asks you to compete against 4 of his students. You have to get the battle grade "Master", at least, to pass. You can fight using the Sword (top option) or Wild Dance (bottom option) fighting styles.
When you win the fight with a battle grade of "Master", at least, the seal on the "Heat Haze Circulation" ability in the Wild Dance sphere grid is removed and you can purchase it with one of your souls.
#6: Dry Leaf Smash, Flying Wild Shot Mastery
After completing #5, talk to Ginryu when you have a 6 star sword equipped (you cannot do the next training with a sword that has a higher or lower star rating).
You have to compete against Ginryu himself and get a battle grade of at least "Master" to pass. You can fight using the Sword (top option) or Wild Dance (bottom option) fighting styles.
When you win the fight with a battle grade of "Master", at least, the seals on the "Dry Leaf Smash" ability in the Sword sphere grid and the "Flying Wild Shot Mastery" ability in the Wild Dance sphere grid are removed and you can purchase them with your souls.
After the fight, Ginryu gives Ryoma 2 Revelations Documents.
To have the Sword fighting style revelation, continuously use the "Beautiful Technique - Weather Vane" ability (press triangle when an enemy attacks you while you are defending). Eventually, after performing the technique, a revelation indicator will appear on the screen (press triangle).
To have the Wild Dance fighting style revelation, continuously use the "Fenghuang Formation" ability (press circle). Eventually, a revelation indicator (press triangle) will appear during the Fenghuang Formation ability.
Sword Fighting Style Limit Break
When you have reached rank 25 of the Sword Fighting Style and have purchased every ability on the Sword sphere grid, speak to Ginryu and fight him.
When you defeat him, you can now level up the Sword Fighting Style past rank 25, up to rank 50. Also, your Sword sphere grid resets - you don't lose the abilities you purchased the first time round, but when you purchase upgrades in the sphere grid the 2nd time round the effects are enhanced (for example, when purchasing a health upgrade you yet again get another health boost, or when you purchase an ability it does more damage, etc).
Wild Dance Fighting Style Limit Break
When you have reached rank 25 of the Wild Dance Fighting Style and have purchased every ability on the Wild Dance sphere grid, speak to Ginryu and fight him.
When you defeat him, you can now level up the Wild Dance Fighting Style past rank 25, up to rank 50. Also, your Wild Dance sphere grid resets - you don't lose the abilities you purchased the first time round, but when you purchase upgrades in the sphere grid the 2nd time round the effects are enhanced (for example, when purchasing a health upgrade you yet again get another health boost, or when you purchase an ability it does more damage, etc).