Bringing hunter bow to +9 should be enough for early to mid game. Hunter saw's ultimate skill (unlocked at +10) pierce def, always hit and hits very hard so it's viable even in late game. Alternatively, you could find Storm Ruler in Mental Ward (the same place you find Hans). Storm ruler unlocked its ultimate skill at +5 and is equally strong as Hunter Saw most of the time. For armor I recommend Armor of the Sun (found in Burning Forest).
You should adapt your rings to the boss fight though rings that increase Str and Spd are always nice to have. Bonus action rings are late game only and are auto-pick.
Equipments for Dps build are pretty flexible but Tank build revolves around Bunker Shield +10 (some fights are borderline impossible without it).
Once confident enough you could go looking for the katana weapon in the Deep Sea areas. It considered one of the best weapons and will stay with you for the rest of the game (even through DLCs)
Solid advice. The only things I can add to that the hunter saw's ultimate skill does double damage if the target is bleeding, which combines really well with Scrape Flesh (unavoidable, 70% chance of inflicting bleeding).
Useful rings early to mid-game are the rings that boost your crit rate: the Hornet Ring (+20% CRT, can be found in the foggy park) and the Ring of the Abyss (+40% CRT, can be found in the Nameless Forest). Crit rate boosts combine really well with Hans' machinegun (+11 attacks if fully upgraded).
The Resurrectionist's Ring (IIRC dropped by one of the nightmare spirits in the Slaughterhouse) is situationally useful. It cancels out crits and instakill attacks. Very few enemies have instakill attacks, but they can be a headache when you do run into them.
And I'd recommend the Knight of Hearts' Ring (10% HP and MP regeneration). Just beat the boss at Castle Frissel, then exchange the Knight of Hearts' soul in the library.