To perform long jumps, use Shizuka and equip Fleck. While in the air, just before you start falling back down again, press Square and the desired direction. This will make Shizuka float forward in the air for a brief moment.
Some folks may mistake Genji for "just a button masher" - this will get you killed in Genji: Days of the Blade. Remember to block when the situation merits it.
Dodging as a dance move
The new dodging mechanic is key when in a crowd, or pretty much whenever you are in a fight. Many times dodging about the battlefield is even more effective than blocking.
Use the correct character for the situation. Each of the four characters have their advantages and you should use them. Many folks will fall in love with a particular character and then won't have that character available in a fight and will be lost without them. Switch up, get familiar with all four. They each are fun in their own way.
Level up weapons!
The Mashogane you pick up after the first act can be used to upgrade your weapons. Find your favorite weapon for each character and keep upgrading it. Also, do not neglect your new weapons. There are five weapons per character, as you get new weapons, check them out and see if you like the way they play. If you do, put them in your weapon swap and start upgrading the new weapons.
When you find a battlefield, especially one that you can go back to (Ichinotani), hang around and get as many kills as you feel comfortable getting (without dying of course). This Mashogane will power up your weapons before you move on to the harder levels. It is not required, but does make life much easier.
Always use kamui
You have a limit to how much Kamui you can 'store up'. When the bar is full, you are 'wasting' Kamui. This is not to say that you should pop it immediately, but don't run around forever waiting for the perfect time to use it. The bar fills up really quickly, so use it often. When I play, whenever I have 3 or more enemies around me and a full Kamui bar, I usually use it. There are exceptions to this rule, when you know (or feel) you are about to run into a boss fight or have just died trying to go through a tough section, store up the Kamui prior to the ugliness and make your life easier. Another thing to note about Kamui, it is character independent, meaning that each of your four characters can all have full Kamui. So if you have a really, really bad fight, store up or use items to cap off all four characters and then use Kamui from each in succession.
You get knocked down....
When you get knocked down, not if, when, press X to get up. If you time it right, just as you hit the ground, you will pop back up to your feet. If you don't time it right, mash the button a few times to get up more quickly and avoid the cheap shot from the AI.
Use Shizuka and equip Fleck when facing many enemies that are grouped together. Then, run towards the enemies and press TRIANGLE while running. Try to hit an enemy with more health than the others. This attack inflicts damage but also plants a bomb on the enemy. Hit the enemy with any other attack to detonate the bomb. The explosion of the bomb does will not damage you.
Use Benkei when fighting a Juggernaut. Equip Benkei's club and hold Triangle until his club glows and sparks. Then, release TRIANGLE and charge the Juggernaut. Make sure that you do not press R1 before holding TRIANGLE.
While in Kamui mode, press CIRCLE if the "CIRCLE" symbol appears instead of any other attack. This will inflict more damage. If the "CIRCLE" symbol does not appear, then choose another button and press CIRCLE afterwards. This will inflict more damage.
After defeating the Giant Crab boss on the beach, move forward to the next area and save the game. Then, go back to the beach where you will find respawning enemies modified with the Mashogane. Continue to kill them to gain Mashogane. When your health is low, retreat back to the save point.