To start, you should have: Full (G) Adamant Armour or Full (G) Rune Armour, Paladin Gloves, Elite Amulet of Strength, 80 / 80 / 80, Sunset Rapier, Auto Eat - Tier III
At the end, you should have: Full (G) Ancient Armour, Paladin Gloves, Elite Amulet of Defence, 88+, Dragonfire Shield, Sandstorm Ring
Elite Amulet of Defence from upgrading Amulets of Defence, which are dropped by Purple Goo Monsters, and Silver Diamond Ring, dropped by Mummies, are our best friends for now; the limiting factor for the remaining dungeons is how much DR you have access to.
If you've only got (G) Adamant Armour, you will need about 920 Hitpoints. If you've got full Rune (G) Armour and Paladin Gloves for 32% DR, 880 Hitpoints instead will allow you to idle Volcanic Cave. Always make sure to use Can I Idle to verify your specific gear setup before committing to any idle adventure. Regardless of which option you choose, the reward for the dungeon is a very useful cape, the Fire Cape, which boasts a very nice 2% DR. If you are just slightly below the threshold needed to idle the dungeon, you can opt to clear it once manually and then use the DR from the Fire Cape to idle afterwards. The only dangerous part is the final boss Malcs, the Guardian of Melvor, who has a very slow and predictable attack pattern and can easily be manual eaten to avoid lethal damage.
At this point, your main sources of food are Raging Horned Elite (higher food/hr, more GP/hr but no prayer or bolts) or Standard Resupplies (if you need prayer or bolts). Use Ancient Sword for farming if you have it, otherwise Sunset Rapier.
Once you have the Fire Cape, you can farm the Sandstorm Ring from Sand Beasts, which will be your best-in-slot Melee DPS ring for awhile. You can even use Safeguard if you'd like to farm it sooner, as it will speed up your Volcanic Cave clears.
Hopefully you find the Volcanic Cave comfortable, as the goal is to get the Ancient Platebody, Ancient Platelegs, Ancient Helmet, and Ancient Shield. There are no Ancient Boots and Dragon Boots are unobtainable on CO (as is the Dragon Helmet), so you'll continue rocking your (G) Rune Boots for quite a while. Do not worry about getting the Dragon Platebody and/or Dragon Platelegs as you will want to save your bars for (G) Ancient armour. You should also start using Prayer obtained from Standard Resupplies to help speed up the dungeon and save food. You should expect to use about 1.5k-4k Trout per hour, depending on how many (G) Ancient pieces you have obtained for additional DR and dps.
Once you get your full Ancient Armour, make another visit to Statues to finish off gathering Gold Bars and Silver Bars and upgrade everything to (G). Luckily the Elite Chests you've been opening also drop plenty of Silver Bars and Gold Bars.
In total, you'll need 6,000 Silver Bars and 10,000 Gold Bars for full (G) Ancient Armour. If you get all the Silver Bars you need, move to Purple Goo Monsters for faster Gold Bars. You can also hold onto the Amulets of Fury for the Fury of the Elemental Zodiacs grind later; you'll need 10 Fury Amulets in total.
After getting the (G) Ancient Shield, there's nothing stopping you from immediately upgrading it again to the Dragonfire Shield, so head to the Green Dragons to get 7,050 Dragon Bones. You can alternatively wait until getting some Ranged levels, as Green Dragons are slightly weaker to Magic and Ranged than to Melee, but at this level, you shouldn't have any trouble, and the 8% DR of the Dragonfire Shield will be very useful as a Magic and Ranged offhand for God dungeons.
Once you have full (G) Ancient Armour and Dragonfire Shield for 43% DR (or 40% with Amulet of Looting and Sandstorm Ring, it's time to grind any remaining melee skills to ideally at least 90+ / 90+ / 90+ while working your way towards Level 80 for your next upgrades from Shrouded Badlands.