Same as with the Cleric Beast, Father Gascoigne is essentially designed to be taken down at base level (eg. Blood Level 4), as he’s the first mandatory boss in the game. There’s no way around him, at least not anymore, so you better get comfortable with fighting him.
For a first boss, Gascoigne is really aggressive and hits quite hard on top of that. If you’re good at taking in your surroundings, you can just fight him the way I did, at the lower level of the stage, on the graveyard. If you find yourself getting stuck on tombstones and getting cornered in the middle of the stage, I’d recommend fighting him on the stairs and the area the stairs lead to, in front of the gate that opens the way to Cathedral Ward.
Unlike the Cleric Beast, Gascoigne can be staggered for visceral attacks, if you interrupt his attacks with a gun shot. You can also backstab him, if you land a charged R2 on his back.
In theory this fight is extremely easy, but his speed and damage potential at this point in the game can lead to quick deaths if you’re not careful. Just use your gun to your heart’s content and memorise the timings, at which point in his attack animations you can shoot and visceral him.
Once he changes forms and turns into a beast in his third phase, the same rules still apply, but you are essentially fighting a completely different boss now, with a whole new set of moves that replaced the old ones entirely. While he doesn’t have access to his blunderbuss anymore, he gains a leap attack, so he can still attack you from afar. The strategy here doesn’t change though; time your gunshots and keep interrupting his attacks and visceral attack where possible.
I opted to bring the tiny music box for this fight, which you obtain from his daughter, to make the third phase a bit more manageable. Same as with molotov cocktails and oil urns, this is completely optional though and by no means necessary. He is very vulnerable to fire though, especially in his third phase, so if he’s giving you trouble, don’t be ashamed to try those.