You see the nodes on the purified Nidhogg picture? Click one, then click the left button and fight the battle. You can do this twice per day per node. You can also get bonus rewards for meeting certain objectives that are listed (like TP above X amount, 2 parties with 650+ speed). You know when you've met an objective when there is red text next to it.
Once you've cleared node battles, click the middle button to send a group of knights on an expedition. These cost 5 stam per hour, and shorter expeditions are more point efficient if you stay on top of them. Knights used here are not locked out of use anywhere else (except maybe other expeditions).
EDIT: Oh, and more nodes/expedition slots are unlocked when you get more points.