One thing to note is that your first aid quests to unlock 225-300 will vary depending on whether you're horde or alliance, not by much, but different NPC's are involved. Any of the required books can, usually, be bought off the Auction House for 3 to 5 gold, saving you the time of travelling to the vendor. The last book, Heavy Frostweave Bandage must be found, it's not available to buy.

Note that once you are past skill 150 you will need to buy books and complete quests to raise your first aid skill. You will be able to buy the grandmaster skill from the Northrend trainer, but the last book you will need to find.

First aid 1 to 150 skill
Materials needed to get to 150 skill
150 - 225 skill
Triage Quest
225 to 300
300 to 375 skill
Beyond 375
Ultimate Professions Guide
1-150
Learn First Aid

Linen Bandages 1 - 40

Train Heavy Linen Bandages

Heavy Linen Bandages 41 - 50

Train Journeyman First Aid

Heavy Linen Bandages 51 - 80

Train Wool Bandages

Wool Bandages 81 - 115

Train Heavy Wool Bandages

Heavy Wool Bandages 116-150

Estimated Materials Required to get to 150 First-aid skill
150 x Linen Cloth
90 x Mageweave Cloth
110 x Netherweave Cloth
250 x Runecloth.
140 x Silk Cloth
125 x Wool Cloth
150-225 First Aid
At 125 to 150 you should get the book. Expert First Aid and two other books.

Alliance: Find Deneb Walker in Stromgarde Keep, Arathi Highlands, Buy the Expert First Aid book. Also buy the Manual: Heavy Silk Bandage and Manual: Mageweave Bandage while you're out here. Or just buy them off the Auction House.

Horde: Buy the books from Balai Lok'Wein, in Brackenwall Village, Dustwallow Marsh.

Train Silk Bandages

Silk Bandages from 151 - 180.

Use Manual: Heavy Silk bandage

Heavy Silk Bandages 181 - 210

Use Manual: Mageweave bandage

Mageweave Bandages 211 - 225

Triage Quest, For First Aid of 225+
For both Horde and Alliance:

Required: You must be Level 35+ and have 225 Skill
Bring a few stacks of Runecloth and Mageweave and you will be able to stay here after completing the quest and powerlevel your First Aid to 300, picking up the new skills at 240, 260, and 290.
Alliance Side:

First talk to Nissa Firestone in Ironforge, she'll send you on your way to Doctor Gustaf VonHowzen in Theramore, Dustwallow Marsh, to do your Triage Quest. It's also quite possible to simply head to the doctor and skip Nissa.

Head to Doctor Gustaf VanHowzen in Theramore, talk to him to start your Triage quest, this will, upon completion, allow you to continue on past 225 up to 300 skill. You'll also have to return at 240, 260, and 290 to train new Bandage types. Consider him your new first aid trainer.

Horde Side:

Talk to Arnok in The Valley of Spirits, located in Orgrimmar, he'll send you on your way to Doctor Gregory Victor in Hammerfall, Arathi Highlands, for your Triage Quest. As with the alliance you can also skip this part and simply head over to the doctor.

Talk to Doctor Gregory Victor in Hammerfall to begin your Triage quest, this will, upon completion, allow you to train up past 225 skill until 300. You'll be required to return at 240,260, and 290 skill to train new bandages. This is your new first aid trainer.

Triage:

The Triage quest for the Horde and the Alliance is basically the same. The only real difference is that it's easier to view all of the Horde beds than it is for the Alliance beds. Everything else is pretty much the same.

Triage will require you to save 15 different patients in a specific order before allowing a maximum of 6 to die. When you begin you'll be in a room surrounded by 6 different beds, stand directly in the center of these beds so as to be able to view them all at the same time.

At this point make sure you're using your special Triage bandages, given to you for this quest, otherwise you'll fail. As to the order of bandaging you'll often find that critically injured patients die unless you start bandaging immediately after they spawn, so you can't just ignore them.

Your general order of operations is to bandage the most injured patient first. Do Critically Injured first, then Badly Injured, and finally just Injured.

One last trick is to target the next patient in line even as you bandage one, this allows a better reaction time to bandaging multiple patients in a row.

Good luck.

225-300
Once you have completed the Triage quest:

Mageweave Bandages 226 - 240

Train Heavy Mageweave Bandages

Heavy Mageweave Bandages 241 - 260

Train Runecloth Bandages

Runecloth Bandages 261 - 290

Train Heavy Runecloth Bandages.

Heavy Runecloth Bandages 291 - 300

300-375
Get the book: Master First Aid - Doctor in the House. You will find it in the Outlands, for about a gold, or on the AH for 5-15 gold.

If you're Horde, get to the Outlands and find Aresella, located at 26,62 at Falcon Watch in Hellfire Peninsula.
If you're Alliance, get to the Outlands and find Burko, located at 22,39 at at the Temple of Telhamat in Hellfire Peninsula.
You will also need to buy the Netherweave and Heavy Netherweave books from these vendors or from the AH.

Heavy Runecloth Bandages 300-330

Netherweave Bandages 330-360

Heavy Netherweave Bandages 360-375

 
First-Aid Beyond 375
Go to Northrend and talk to the guard to find the first aid trainer for your town. You will get Grandmaster First Aid and learn Frostweave Bandages. These will take you to at least 400 skill.

To get to the 450 first aid skill you will need the get the Heavy FrostWeave Bandage manual. You will need at least a 390 First Aid skill and you will have to go to Zul'drak in Northrend and kill mobs until one drops. Mobs there are about level 74 to 75. I found mine on a Drakuru Prophet. You do not need to enter the instances, just kill mobs in the general area.

They manuals are Bind on Pickup, so you won't see them on the Auction house and you can't get them from a vendor.

Your Heavy Frostweave Bandage skill will go green at 430, so you are going to make a lot of bandages if you want to max out your skill.