Your skills and stats are leveled by simply using them. If you want to level your swords skill, swing a sword at a monster. You can change your starter melee weapon type by visiting any merchant NPC (Councillor, Mage, Fighter, etc.) and it costs 20 gp to do so. Using a starter weapon to skill will take you only so far. At some point you need to start using a better weapon to get better at handling that weapon type.
All weapon skills, besides the basic ones that player starts with, can be leveled by consuming skill books. These can be crafted with the Scribing skill or bought from other players. The Manual Page ingredient for the skill books can be obtained as loot from warrior type monsters or from other players via trading.