If you want to capture an NPC's soul, do so where no one can see you, as casting a soul trap on someone will incur a bounty if spotted. When you become better acquainted with capturing souls, consider tackling larger targets such as Daedroth, Lich and Gloom Wraiths, which have high-leveled souls. Creatures such as deer, goblins, mud crabs and rats are easy to practice using Soul Trap spells on. The easier the creature is to kill, the smaller its soul level will likely be and the smaller the soul gem you'll need. Locate a creature who's soul you want to capture, or if you have a Black soul gem, an NPC (such as an Arena combatant or bandit).
In addition, you can obtain Azura's Star, a re-usable Grand soul gem artifact, by completing the Azura daedric quest.
Soul gems can be bought from most mages stores, in random loot and from Mages Guild halls and range in price depending on their level. Black soul gems are technically like grand soul gems, but can capture NPC souls as well - they can be created using methods discovered during the Necromancer's Moon Mages Guild quest. For example, a mud crab with a soul level of 'petty' can fit in any size of soul gem, but a Minotaur Lord's 'grand' soul level can only be trapped in a grand or black soul gem. Although any soul gem can be used to capture a soul, the larger the soul, the larger the soul gem needed.