

Fish Tank
The Bigger the better?
- Size: 20 Gallons for the first axolotl and 10 gallons each additional axolotl. Axolotls are ambush predators and may occasionally attack tank mates. Look for a wider tank instead of a taller one. This is the piece most likely to be found used. You will notice that tanks get pretty big, the reason for this from a hobbyist perspective is the larger the tank the harder is is to change water parameters, which means a more stable tank.
- Substrate: Substrate is what you use to cover the bottom of the tank. Sand makes it easier to keep the nitrogen cycle even but many keep the tank bare or use large river rocks, lego floors, astroturf, tile, and many other options. One rule, though:
No gravel! Axolotls will attempt to eat the gravel which can lead to intense pain and death.

Filtration
What's under the hood
No one is going to visit your house and say, "Hey tell me more about your canister filter." The truth is the right filtration system for your needs is the difference between enjoying your tank and wishing you had never heard of aquariums.
HOB filter: This means "Hang On the Back." The advantage here is that it is less expensive and it is easier to give the charcoal the occasional rinse. The disadvantage is that it tends to have less filtration power and the biological media isn't as efficient.
Canister Filter: Advantages: Quieter and more efficient. Most also have flow control so you can create that smooth, pondlike condition usually loved so well.
Both? I have both on my kid's tanks because that way I KNOW there's a filter even if I'm not in there every day checking on it. We run the output from the chiller through the HOB filter instead of using the filter's pump.
In either case make sure you have twice the power than the gallons of your tank. Your Axolotl is going to poop. A LOT.

Test Kit
Your neighborhood pet store probably tests water for free! However, when you are setting up a new tank, you will need to test your water many times, so it is more convenient to have the test kit on hand.
I recommend the API freshwater test kit. It is very accurate and will cover testing all the parameters you need.
The API Freshwater test kit is a common language among aquarists, so if you need expert help you can be ready to provide tank details.

Temperature
I thought that since axolotls live in Mexico that they would prefer tropical weather. The truth is they prefer to be cold. 65 degrees is a good temp, 70 is too hot, and 72 will be fatal.
The aquarium will naturally be the same temperature as the room it is in. So, if you keep your home at 65 degrees half the battle is already won. Otherwise, here are three ways people use to cool their tanks:
1) Frozen water bottles. You place the frozen water bottle entirely in the tank, when the contents of the bottle melts, you take the whole water bottle out of the tank. This is great if your small tank dips a few degrees high for a few weeks in the summer. Not stable, but cheap and it does work in a pinch.
2) An aquarium fan. Great solution if you need to drop the temperature of the tank. For some the noise is an issue, for others it is a white noise maker that lulls them to sleep. The good thing to know if that the fan works through what is called "evaporative cooling," which means you will need to top off your tank more often as it is running. The less humid your climate, the more effective the fan will be and the more you will have to refill the water. Having an air bubbler will also make this more efficient as it breaks down the surface tension.
3) An aquarium chiller. You will pay around $250 for a chiller. The advantages are that it is reliable, silent, and precise. the disadvantage is the price tag.
Don't put ice in the tank directly. Tapwater typically has chlorine in it, and you need to use water conditioner for any water you add to the tank.

...And Everything Else
Your axolotl will need a hide. A place to sleep. When you get a baby most aquarium hides will do. As they get older, they won't fit in them anymore. Clay pots, Pvc pipes, carefully stacked rocks are all options. Remember, your axolotl doesn't have eyelids, and needs to go someplace dark to chill.
Speaking of someplace dark, it is easy to over-purchase in the lighting department. A clip-on light will work fine. You will want something for taking pictures, showing friends, and checking your axolotl's health.
An air pump and bubbler stone are cheap ways to provide your axolotl with entertainment and break up the surface tension in the tank. You don't want to create too much of a current in the tank, so keep the bubbler on the opposite side of your filter's intake and keep it small so the axolotl can choose when they want to interact with it.