This is a suggested route through the game, you can leave most bosses until you have all the major upgrades. For sword / armor upgrade you only need to beat the 1st boss and the dual wielding miniboss in Dark Forest Area-2. The Frozen Mines miniboss is weak to Spears, they pierce its armor.

Town of Glenfinnan - Axe
Dark Forest Area-1 - Torch
Dark Forest Area-1 - Siren Scale
Dark Forest Area-1 - Spear
Underground Temple - Golem Armlet
Underground Temple - 1st Armor Mark (secret wall after Golem Armlet)
Dark Forest Area-2 - Combat Armor (secret wall after miniboss, tough platforming)
Dark Forest Area-2 - Steel Claymore (secret wall before boss screen)

It's advisable to rush the remaining relics, you can do so without needing to kill bosses.

Underground Temple - Pazuzu Cape (after miniboss)
Devil's Peak - Harpy Boots (snow section after skull gate, middle moving platform)
Devil's Peak - 2nd Armor Mark (secret wall in locked door vertical screen)
Aqueducts Area-1 - Fenrir Ring (need double jump, underwater snake section)

Frozen mines has the armor first but getting the sword makes the armor fight a lot easier.

Frozen Mines - 3rd Armor Mark (need double jump, after 1st checkpoint above minecart)
Frozen Mines - Silver Claymore (gate puzzle before boss)

Most runestone / heart / weapon bag pickups are easy to find but there's one secret in Dark Forest Area-2 that's very elusive, I'll list the location here as an exception.

Dark Forest Area-2 - after the locked gate, in the underwater section with electric vines, there are floor spikes placed on top of breakable tiles, weapon bag is there. It's right before the moving platforms.