What are the major things one needs to know to start Dark Souls 2: Scholar of the First Sin?

If I recall correctly the tutorial mainly consisted of teaching you the controls. If you have those figured out you haven't missed much. The full control list can be found at the wiki here.

Upgrading Equipment

You've learned how to level up so the next most important thing is learning how to upgrade your equipment. As you explore you will encounter several blacksmiths who will give you the option to upgrade your equipment when you talk to them. Upgrading equipment will cost souls as well as upgrade materials. Different weapons require different materials to upgrade, some materials can easily be bought from the smiths that upgrade your weapons and some must be found. Each weapon only has one upgrade path unlike in past Souls games.

Eventually, you will find a blacksmith who will be able to infuse weapons when you give him a key item. Infusion just means adding an effect to your weapon such as fire, lightning, magic, or poison. This infusion upgrade is independent of upgrading a weapons level. So infusing a +4 long sword with lightning will result in a +4 lightning long sword. Materials for infusion can be found or bought in limited quantities.

NPCs

On your travels you will encounter several NPCs. You should talk to them. However, they will not reveal everything they have to say at once. Talk to NPCs multiple times until they start to repeat themselves. Once the dialogue of NPCs is exhausted they will often be of some benefit to the player. Some give you items, others will sell you items, some even move into Majula for easier access.

Messages

You will often see orange mark on the ground. If you interact with them they will bring up a message left by another player. Some player messages are useful warning you of danger to come or giving hints about secrets in the area. However, some players leave messages to mislead you so always exercise caution. You too should leave messages for other players by pushing start and navigating to the message menu. Share your knowledge, or cause a bit of mischief. Or leave funny messages in front of skeletons.

Finally

This should be enough to get you started. Always remember that the wiki is your friend if you are confused about something. Dark Souls 2 really likes to withhold information (hidden effects on some equipment, random walls open up to secret rooms, etc.) so there's no shame in using outside resources.

Dodge. Dodge a lot. Enemies telegraph their attacks well ahead of making them. This isn't to tell you when to dodge, but where. Don't be in the spot your enemy is going to be hitting.

Don't use a shield to block anything from anyone bigger than you. If they're taller than you, you're just going to get knocked down. Also, don't turtle attacks too much.

Dark Souls is largely a game of movement and positioning. Superior positioning is key. A naked man with a stick is more dangerous than the Pursuer, as long as he's never where you swing, and he's always hitting you. Never be still for long.

Keep an eye out for any advantages the environment can give you. Undead are mindless, and will follow you into narrow spots where they lose the advantage of numbers. Their ambushes will fall to rot if you funnel them all over a bridge that you smack them off of, or retreat to a wide open place that gives you plenty of room to move around.

You'll also hear some complaints about "Dudes in armor". Understand that anyone in armor follows the same movement rules you do. Four legged enemies and other behemoths do not. Duels to the death, mano a mano is more Dark Souls II style than any other Souls title.

That's as much as I can say about it right now. Have fun, and don't give up! There's a way through everything.