This is a scaled down (TLDR) blueprint of precisely what you need to do to execute the POE1 campaign effectively. I have in mind newer players and especially those who come from POE2 in making this.
The POE1 campaign can be completely overwhelming for a new player (which was a significant reason for creating POE2). If you are new, follow this guide zone for zone, hang in there, and buckle up for the ride.
Some decisions I've made:
Each "area" is treated as its own "zone" to avoid confusion about what constitutes an official zone and what does not.
If a zone can be skipped because it ultimately offers no real value, it is skipped in this guide.
If the progression of the game generally suggests doing something as you go (skill books, ascendancy, etc.), this guide will call it out as you go.
You wake up on the beach, grab obvious weapons and gems on your way up the coast. You will approach your first boss: HILLOCK.
A simple pattern of attack + move (to dodge) will take him down easily. You can spam your potions to empty because the town is next.
Zone 2 - Lioneye's Watch (A1 Town)
There are 3 vendors of note:
1 - Nessa - Generally potions, jewelry, and caster items
2 - Tarkleigh - Generally standard weapons and armour
3 - Bestel - Gambler and eventually respecialization
Your first time through, grab your first skill gem from Tarkleigh before heading up to The Coast on the North side.
Zone 3 - The Coast
You are just going to sprint through this Zone and go straight to The Mud Flats. You are welcome to slay some monsters but the XP gain here is quite small compared to the flats.
It is generally considered best to:
Grab the Waypoint
Skip Tidal Island (for now) -- really, just SKIP IT
Proceed to the Mud Flats
And eventually you will take the Submerged Passage waypoint back to Tidal Island
Zone 4 - The Mud Flats
Skip Fetid Pool altogether.
Here is your first quest (Breaking Some Eggs) you have to complete. Slay the Rhoa, Grab 3 Eggs, and Proceed to the far side of the zone where you'll put the eggs in the wall to open The Submerged Passage.
Quest Reward: Level 4 Gems
*Note that for a new player the Rhoa monsters can be really gnarly. Generally best to strafe them with attack + move, and move laterally because they have a point-to-point charge move that stuns you.
Zone 5 - The Submerged Passage
Your first time here you are going to grab the Waypoint and the head to town. Talk to some vendors + buy/sell any gear you need. You will then take the town waypoint back to The Coast, and proceed to Tidal Island (which is near The Coast Waypoint).
Zone 6 - Tidal Island
Here you complete the quest Mercy Mission by defeating HAILRAKE at the bottom end of the zone. Hailrake can be an annoying fight as he intermittently casts frost/chill effects. Again, attack + strife intermittent tactics is highly recommended.
Quest Reward: Quicksilver Flask (and Gem Reward)
After defeating Hailrake, return to town (you can do this by exiting to character selection and re-selecting your character). Collect rewards and then waypoint back to The Submerged Passage.
Zone 5 (again) - The Submerged Passage
We are back, this time you will follow the paths until you land in either the hole in the ground to The Flooded Depths or the next zone The Ledge.
A common strategy is to leave a Town Portal at the entrance to the Flooded Depths, then proceed to The Ledge and all the way to the Lower Prison, returning to the Depths to finish the The Dweller of the Deep. This is because the portal / waypoint progression of doing things in sequence is inefficient. This is also in part because the Dweller of the Deep can be a very challenging fight.
The progression of this guide will cover doing these zones IN ORDER.
Zone 7 - The Flooded Depths
Search this zone until you find The Dweller of the Deep boss. Defeat the Dweller for your first Skill book reward as part of the quest of the same name: The Dweller of the Deep.
Quest Reward: Passive Skill Point
If you have progressed the zones in order, proceed to The Ledge (through The Submerged Passage)
Zone 8 - The Ledge
This is a walkthrough zone with an optional boss. Progress linear-fashion through this area, grabbing the Waypoint in the middle. (Note that just after the waypoint there are three stones arranged in a triangle which points the correct way to the zone exit).
Zone 9 - The Climb
This is another walkthrough zone with two optional bosses. There is a Waypoint near the start (a bit of the way forward off to the side depending on your starting orientation). At a far end of the zone you will see a wall door that takes you to The Lower Prison.
Zone 10 - The Lower Prison
When you reach this zone you have completed the first part of The Caged Brute quest, and you can return to town via the Waypoint at the start to shop the Level 8 Gems and vendor any gear upgrades that will assist with the next few zones.
Quest Reward: Level 8 Gems
This Zone contains your first Trial of Ascendancy. If it is your first time through the campaign these are good to do while IN the zone (though if you miss one you can come back and complete it later).
After finding and completing the Trial, proceed to The Upper Prison.
Zone 11 - The Upper Prison
Important to mention that this zone has a hidden stash of Flasks, that can be very helpful for an efficient progression. You want to look for a Strongbox icon that appears unreachable (behind a wall). And near that wall on a pillar you will find a "SWITCH". Click this to open the wall, and complete the strongbox for a random roll of flasks:
After collecting the flasks, search for the entrance to The Warden's Quarters (which is technically still a part of The Upper Prison).
Zone 12 - The Warden's Quarters
Here lives the infamous Brutus. Good luck.
Upon entering the zone, follow the long line-path to its end, then turn and backtrack the same length on a parallel path back, and then go up through the small rooms. You will find Brutus at the end.
For new players, the most common need for Brutus is SUSTAIN, and know as well that unlike POE2, when you die and come back in the zone, Brutus does not entirely respawn, so you can continue the progression of the fight straight through dying as many times as you need.
Quest Reward: Level 10 Gems
After defeating Brutus, proceed through to The Prisoner's Gate.
Zone 13 - The Prisoner's Gate
You will see the Waypoint immediately at the start of the zone. This is a walkthrough zone that has a fake exit at the end of that actual path which will be closed by Piety. After you run into this, try to search the opposite area near the starting point of the zone. Hug the wall and follow the path that seems to be taking you "deeper". This search can feel a bit dense at first, persevere. You are looking for the Ship Graveyard.
Zone 14 - The Ship Graveyard
After grabbing the Waypoint at the start, you are looking for a cave-entrance area called The ShipGraveyard Cave. It will appear somewhere more central in the zone, near a sunken/broken ship. Don't get distracted by other zones / points of interest, you just need to find this Graveyard Cave hole.
Zone 15 - The Ship Graveyard Cave
Proceed in a linear fashion directly away from the zone entrance until you come upon the boss Stranglecharm, defeat her and grab the Allflame quest item. There is a wall exit behind her that takes you back up to The Ship Graveyard.
Zone 14 (again) - The Ship Graveyard
You will come up from the Cave right in front of Fairgraves. Defeat him and his armies to finish The Marooned Mariner quest, which will give you another Skill book when you get to town.
Quest reward: Passive Skillpoint
After defeating him, return to the The Ship Graveyard zone (NOT the Cave) to find the entrance to the Cavern of Wraith. This entrance will be on a wall.
Zone 16 - The Cavern of Wraith
This zone has a Waypoint right at the start, and from there this is just a walkthrough zone that eventually brings you to the Cavern of Anger.
Zone 17 - The Cavern of Anger
Another walkthrough zone where you are going to the far end of the zone to find a wall entrance to Merveil's Lair.
Zone 18 - Merveil's Lair
You made it to the A1 Boss, Merveil. Good luck.
She deals significant cold damage and has several phases. Try to avoid the large cyclones that stretch to the sky, and wipe out any minions she spawns. Defeating her opens the way to Act 2 (at the North side of her cave) through The Southern Forest.