There are a myriad of ways to change your pet's appearance on the Neopets site. The primary changes available are color and species but there are many options available for achieving either. It is important to note that any item which changes species will also change the color of your pet. Therefore, I will adress color changes first.

Changing color without changing species

Paint Brush
Paint brushes are the most obvious and well known way to change a pet's color. Below are some pieces of trivia about Paint Brushes:

Retired paintbrushes will still work! Only retired colors no longer work.
There are only two retired colors: Stone and Glass. Pets that were painted Stone and Glass turned into Mystery Island and Tyrannian versions, respectively. (Credit to Neocolours for this information.)
Currently only three Paint Brushes may be bought. Baby, Royal, and Darigan are all available for purchase in the Hidden Tower in the Faerie City.
Paintbrushes, even retired ones, may be gained through a Random Event or won from the Fruit Machine in the Lost Desert, the Alien Aisha Vending Machine in Neopia Central, or by completing a World Challenge map.
Halloween Paint Brushes may also be won from Test Your Strength.
Tyrannian Paint Brushes may also be won from the Wheel of Monotony.
Here are the available Paint Brushes.
Starter Paint Brushes are available in the Newbie Pack of a brand new account but these items may not be sold or traded. They only come in the basic colors: Blue, Green, Red, and Yellow.

Chia Pops
Less well known than paint brushes are the magical properties of certain Chia Pops when fed to a Chia. This is the sure-fire way to get a fruity Chia.

Chia Pops may be bought from the Snow Foods store on Terror Mountain.
All of these items are r99 which makes them very rare. It also makes them gourmet foods and highly sought after. These items will always be very expensive if bought from a user shop.
Chia Pops which will make your Chia into a fruit or vegetable.

Fountain Faerie
The Fountain Faerie will give out quests just like the regular faeries but she is incredibly rare. If you are lucky enough to find the item she is looking for then you may afterward visit her Rainbow Fountain in the Faerie City and paint a pet of your choosing. Below is some further information about the quest and paint options:

Which item she asks for can vary widely in rarity and price and seems to come from any item group. Some people get away with paying only a few thousand and others are asked for an item worth several hundred thousand.
It is commonly agreed that Royal, Robot, and MSP are unavailable from the Fountain. TNT has confirmed in an Editorial that Ice is unavailable from the Fountain Faerie.
Once you complete the quest, you may go back at any time and paint any pet you have on the lucky account. It does not matter which pet is active when you complete the quest and your free ticket into the Rainbow Fountain will not expire.

Random Events
Yes, your pet can actually change color as the result of a Random Event. As far as I know, these are all rather rare.

Red: If your pet becomes angry there is a slim chance that it will turn Red. This will most likely occur to pets which are hungry.
Blue: If your pet becomes sad there is a slim chance that it will turn Blue. Sick pets will become sad at an alarming rate.
Invisible: The text of the Random Event is: "Oh no! Where on earth is [INSERT PET's NAME HERE], you cant find her anywhere!!! Where on Neopia could she be?"
Baby: A Baby Bruce named Boochi is known to take potshots at pets. Sometimes he will miss but if he doesn't then you get a Baby pet.
The Red and Blue Random Events may target any one of your pets, whether it is active or not.
It is believed that the Boochi and Invisible Random Events only target your active pet.

Changing species
This is where most of the fun is.

Morphing potions
Morphing potions are the most direct way to change a pet's species. Most species morphing potions are available in several colors. Below are a few facts about morphing potions:

If you give a morphing potion to a pet both its color and species will change. For instance, if I give a Green Uni potion to my Blue Lupe tthen my Blue Lupe will become a Green Uni. If I give the Green Uni potion to a Blue Uni, then the Blue Uni will become a Green Uni.
Morphing Potions can be bought from Kauvara's Magic Shop.
The Stone Lupe Morphing Potion exists but will do nothing because the color has been retired.
Here are all of the available morphing potions.

Transmogrification Potions
Transmogrification Potions are a special kind of Morphing Potion. They work just like the Morphing Potion but they will only make Mutant colored pets.

Transmogrification Potions may only be obtained through a Random Event where Dr. Frank Sloth gives you one.
If I give a Lupe Transmogrification Potion to a Blue Lupe, it will turn into a Mutant Lupe. If I give a Lupe Transmogrification Potion to a Blue Uni, it will also turn into a Mutant Lupe.
Here are the available Transmogrification Potions.

Magical Plushies and Toys
These items are very rare. They act in a similar fashion to Morphing Potions.

Magical Plushies and Magical Toys work exactly the same way and do the same thing. To activate this item, you must have your pet play with it.
After the plushie or toy has been played with, it turns into a regular toy or plushie and loses all magical properties.
Some of these were given as prizes in some games very long ago. Now you can only buy them from the Toy Shop in Neopia Central if you are very lucky.
Here are the Magical Plushies. Note: Ignore the non-magical plushies. They don't do anything.
Here are the Magical Toys. Note: Ignore the non-magical toys. They don't do anything.

The Lab Ray
This very special thing allows you almost unlimited access to both color and species changes. As such, I feel it ought to get its own category.

The Laboratory can be accessed by completing the Secret Laboratory Map. You will need all of these pieces in your inventory and then you will need to visit this page in the Neopets Treasure Hunt.
After a map is completed all pieces will dissappear from your inventory. You will not be able to complete the map again, but you will always have access to the Lab Ray through that page.
You may zap one pet per day with the Lab Ray on any account where you have completed the map.
The pieces of the Lab Map may only be obtained through Random Event.
There are many colors which are only available through the Lab Ray (or the Fountain Faerie.) These colors are: Alien, Chocolate, Clay, Custard, Garlic, Ice, Jelly, Marshmallow, MSP, Robot, Snot, Sponge, and Coconut.
There are a few colors which are generally agreed to be unavailable from the Lab. These colors are: Royal, Usuki, Quiguki, Baby, Pirate, and Plushie.
It is believed that all pet species are available from the Lab Ray but Krawk and Draik pets are extremely rare.
The Ice Bori is unavailable from the Lab Ray (or the Fountain Faerie.)
When the Lab Ray changes a pet's species, a new color is chosen at random from the four basic color options (Blue, Green, Red, and Yellow.)

New category! Miscellaneous Items
It seems there are several other ways to change your pet that are best grouped by item type.

Neggs
These Neggs may be bought from other users or purchased from the Neggery on Terror Mountain with Negg points. (To get Negg points, you must trade in other Neggs.)
Thanks to ansky225 for telling me about the neggs!

Kaleideonegg: Possible effects are similar to the Lab Ray. Unknown which colors or species are available.
Plaid Negg: May change pet color randomly. Unknown which colors are available.
Vortex Negg: May change pet color to one of the four basic colors (Red, Blue, Yellow, or Green)

Food Items
Not all of these are confirmed. Try at your own risk.

Chocolate items: May change pet's color to Chocolate. The Neopian Times Editorial Issue #180 touches on Chocolate.
Garlic items: May change pet's color to Garlic. The Neopian Times Editorial Issue #170 touches on Garlic.
Glowing Jelly: May change pet's color to Glowing
Thanks goes to nacey for confirming the Glowing Jelly color change. (You can read about it here .)

Fishing Items
These few items have very random properties. This information is taken (with permission) from Crash_Bandicoot_Psx's petpage.

Flash of Rainbow Fountain Water: Will change the color of the pet it is fed to. The color change is random and, I assume, limited to the colors available at the Rainbow Fountain.
Mysterious Swirly Potion: Will change your pet into a randomly chosen limited edition pet! Kiko, Koi, Tonu, and Cybunny are known possible outcomes. (Other possiblities not confirmed.) The pet's original color will be retained as long as the limited edition pet has that color as well. Otherwise, it will be recolored to one of the four basic colors.

Rumor: Wheel of Monotony may give out other paint brushes after a Tyrannian Paint Brush has been won. Thanks to elwing_elfje for this!

Neopian Times Editorial, Issue 235