Find spirits and their maps then pick paths that will enable you to ultimately identify their specific path. Also use gear and bombs. Make use of leveling up rewards to raise your main acceleration as well as stealth.
When upgrading your attributes or equipment, choose grip until you've finished leveling it. The spirits are pretty tough to fight but tools to catch them in the first place are pretty easy to come by (speed sandwiches and stealth potions).
Although most are in locked boxes either in the four areas or within houses in those four areas, there are a few in additional locations: one is down on the beach, the other is in the spaces or corridoors between locations (try the mailman's house between Serfside and Splitwood). One area to check is behind the statues that carry the fish god's eyes in Rust Grove. Other than that, bash down every locked door and take the opportunity to walk around (rather than run) and see the areas you missed while chasing spirits.