Start whenever you want, but you will have to go to all three major cities of your faction to do seasonal/daily quests  :

• Stormwind (outside, gates), Ironforge (outside, gates), Darnassus (inside, by dummies)
• Undercity (inside, courtyard), Orgrimmar (outside, gates), Thunder Bluff (outside, eleveator)

Pilgrim's Bounty table at Orgrimmar

The other two cities (Exodar, Silvermoon) are needed only for achievement (seating at opposite faction tables) - whereas you will get PVP flagged only in Undercity/Thunder Bluff and Darnassus/Exodar).

First of all, purchase [Bountiful Cookbook]

Buy it from festival vendor to get five recipes you will need (cooking 330+ needed). And be warned, you will need to cook them in various quantities (first time 5, then 20, then some combinations - so stuff ingredient before, as some of them can be purchased only on few places).

• Sweet potatoes - Darnassus / Thunder Bluff
• Cranberries - Ironforge / Orgrimmar
• Pumpkins - Stormwind / Undercity
• Wild Turkey - hunt them in Elwynn Forest / Trisfal Glades
• Spice Bread ingredients - any vendor
• Honey - any vendor
• Autumnal Herbs - any vendor

In fact you will have to visit all cities to get the ingredients.

Pilgrim's Peril (Ally) and Pilgrim's Peril (Horde)

Pilgrim Clothing is something you will need for achievement - 5 cooking dailies will get you all needed. Wear it at least ONE piece), sit at tables of opposite faction cities (all four of them). Die few times. And you are done.

Pilgrim clothing on my Horde shammy

The Turkinator (Ally) and The Turkinator (Horde)

Turkey Shooter achivement is "easy" - just shoot 40 turkey in given time and it's done. Every turkey (Elwynn, Trisfal) will give you "buff" that allows you to extend timer for 30 sec (and also shows how far are you in achieving the goal). Make a macro - /tar Wild Turkey. You can add a /cast of your instant spell. And just smash it while running. And do not try to do together with needed 25 turkey meats for cooking quest.

Quests needed for achievement

Even the tables shows 20 items of each, bear in mid, that there are also quests with another 5 items. So in total, you will need more (25 or 30, look bellow) to cook.

Hint for getting them done as daily - start in Stormwind, collect 20 (40/60) turkeys and get Ripe Elwynn Pumpkin(20-40-60 pieces) and two quests. Then head to Ironforge, get loads of Tangy Wetland Cranberrie (and of course other 2 quests). Next stop is Darnassus, last quest to get and Teldrassil Sweet Potato (20-40/60 pieces). And you can start cooking and finishing Darnassus quest, then either go IF or SW and finish the rest of quests. Always keep excess of cooking mats for next day, so you save yourself one trip.

Pilgrim's Paunch (Ally) and Pilgrim's Paunch (Horde)

This achievement needs Spirt of Sharing buff from all four capital cities - eat all five foods at tables.

Eating at tables to get "spirit" buff can be a bit tricky, as some tables are bugged. You have to the 5 different "foods" from table. So just go from table to table. No hurry. And Spirt of Sharing is nice buff - +10% reputation!

FOOD FIGHT! (Ally) and FOOD FIGHT! (Horde)

Involves tables again (yes, bugged) - take a seat, target fellow player at table, use the "1" option ("pass" a dish).

Sharing is Caring (Ally) and Sharing is Caring (Horde)

Again the tables, you have to "pass" five different dishes. Try different tables until you are done.

Now We're Cookin' (Ally) and Now We're Cookin' (Horde)

Just cook all five different holiday food and achievement is yours. As you will do daily, you will cook lot of them anyway. Remember, Cookiing 330+ required.

Pilgrim's Progress (Ally) and Pilgrim's Progress (Horde)

Achievement Title :) what else - just complete all five holida's daily quests. One day is enough for completing them all.

Terokkar Turkey Time (Ally) and Terokkar Turkey Time (Horde)

Talong King Ikiess awaits in Sethekk Halls (Auchindoun, last boss) on normal and heroics. Wear two pieces from Pilgrim Clothing set and slay him.

Turkey Lurkey (Alliance) and Turkey Lurkey (Horde)

One from quests will give you [Turkey Shooter] which you ahve to shoot rogues (oh yes, rofl) - Blood Elf, Gnome, Night Elf, Troll, Dwarf, Human, Orcc and Undead.

Bountiful Basket and Bountiful Feast

Bountiful Basket can be purchased from festival vendor - combaines festival foods into Bountiful Feast (80AP, 47SP, 40 STA for your party). As any other festival items, this has only 7 days duration!

Turkey Caller

Turkey Caller is reward from quest Slow-roasted Turkey (Ally, Darnassus) / Slow-roasted Turkey (Horde, Orgrimmar) (requires 5 Slow-Roasted Turkey to cook). Funny item with 2 mins cooldown. And yes, poor Lonely Turkey :)