You are free to visit the Bars in any order you like but this is how I did it. It assumes quite a few quests finished though and/or the levels to Rune to places. Here is an overview of how to get to all the differnt places. Depending on your means of transportation, You can do the different Bars in any order you wish, but I found this route the easiest to get them all done.
Jolly Boar Inn
Rune to Varrock and go out through the east gate. Run north to the Jolly Boar Inn.Buy a ‘Olde Suspiciouse’ for 10 rs gold.Attack, Defence, Magic, Crafting & Mining can be affected.
Blue Moon Inn
Go back inside Varrock.The Blue Moon Inn is located in the south of Varrock, right south of the Market Square.Buy an ‘Uncle Humphrey’s Gutrot’ for 50 Gold Coins.Hitpoints, Attack, Defence, Strength & Smithing can be affected by this drink.
Blurberry’s Bar
From there I went to the Grand Exchange and took the Spirit Tree in the north-eastern corner to go to the Grand Tree.Blurberry’s Bar can be found up in the Grand Tree on the first floor to the east.Buy a ‘Fire Toad Blast’ for 10 Gold Coins.This drink will hit you 3 hp.
The Forrester’s Arms
Next I runed to Camelot and from there ran west to get to Seers.The Forrester’s Inn is located to the north-west.Buy a ‘Liverbane Ale’ for 18 Gold Coins.Attack, Defence, Woodcutting, Firemaking & Fletching are affected.
Rising Sun Inn
From there I runed to Falador.The Rising Sun Inn is situated right east of the western Bank in Falador.Buy a ‘Hand of Death Cocktail’ for 70 Gold Coins.Hitpoints, Attack, Fishing & Ranged are affected.
Rusty Anchor Inn
From there I ran to Port Sarim, going out of Falador through the south gate and just running straight south.The Rusty Anchor Inn is situated to the north-east.Buy a ‘Black Skull Ale’ for 8 Gold Coins.This is a safe one to drink :). It doesn’t do you any damage at all.
Karamja Spirit Bar
From Port Sarim I went to Musa Point using my Amulet of Glory. If you don’t have one, you can take the boat to Point Musa from here for 30 Gold Coins.Buy an ‘Ape Bite Liqueur’ for 7 Gold Coins.And another safe one, no damage taken.
Dead Man’s Chest
From here I ran west, round the volcano along the northern coast, to reach Brimhaven.The Dead Man’s Chest is located in the south part of Brimhaven.Buy a ‘Supergrog’ for 15 Gold Coins.Attack, Defence, Prayer, Herblore & Cooking are affected.
Flying Horse Inn
I ran to East Ardougne, but you could also take the boat from Brimhaven to get to Ardougne for 30 Gold Coins.The Flying Horse Inn is located just north of the Castle in East Ardougne.Buy a ‘Heart Stopper’ for 8 Gold Coins.Be prepared to take 17 hp damage!
Dragon Inn
And the last stop is Yanille. I runed to the Watchtower, but you could also walk from Ardougne going straight south, past the Monastry and the Fight Arena to reach Yanille.The Bar is located to the north-west in town.Buy a ‘Fire Brandy’ for 12 Runescape Gold.Attack and Defence will be affected greatly.