To use spells, you have to equip them in your Spell slots. There will be a menu at the Sites of grace. If you started as a samurai, you don't have spell immediately. You should fine them or buy them at trainers.
You can find one at the Waypoint ruins here :
You'll need to meet the stats requirement for spells to be able to use it. For Sorcery it's usually Intelligence.
Then there is incancations, for which you need a seal to cast and usually need Faith but there is different schools which have different Stats requirements.
So basically :
- Equip the spell you want in your spell slots at a site of grace.
- Use a staff to cast sorcery or a seal to cast incantation.
Also if you find your damage too low, level up the stats which scale with your weapons. As a samurai you have a katana and a longbow at first which scale with Dexterity.
In addition, weapons upgrade is very important for upping your damage.