Hi there, I read the full original post but didn't read the replies so pardon me if any of this is already addressed and such. This will be a long post full of lots of information, a lot of which you probably don't need, some of which you probably already know.
First thing I'll talk about is a list of things you will need, the essentials. I'm sure you already know about a lot of it or most of it but missing any of it can be a big deal so I figure it's best to just say it.
1: Freshwater testing kit for ammonia, nitrates, and nitrites.
2: A small fan to keep the water cool. More on this later.
3: A quality filter with quality media. More on this later.
4: No rocks! More on this later.
5: Dechlorinator, which you have.
6: Hides or densely planted area for the axolotl to get away from lights and such.
7: A full grown axolotl will need at least a 20 gallon tank (pref. 20 long, 30x12x12)
Fan
As for the fan, I like to get what's called a "usb stand", as well as a "usb adapter" and a usb fan. Obviously you don't need to do it exactly like this, but it's cheap, looks good, and it's quiet. The usb stand is basically just a usb plug in a plastic housing with a long cable, you plug that long cable into a usb -> wall adapter, plug that into the wall, and the fan into the upper usb stand. I used double sided tape to attach the usb stand to the rim of the tank and then some black duct tape on the outside to ensure it wouldn't fall into the tank and such. It cools by evaporation more than anything else, by constantly blowing on the surface of the water it will create faster evaporation which cools the tank, scientific wizardry stuff that I don't really understand, but it works amazingly well. I had a 50 gallon tank that stayed 60 degrees year round with just one of these fans. Here are links to the specific things I used, like I said you could change it up however you prefer. One fan will easily keep most tanks below 70 degrees, if you get up to like 75 gallon+ tanks then probably add another fan or two.
Amazon.com: ARCTIC Breeze Mobile, USB-Powered 92mm Portable Fan, Portable Cooling Solution, Quiet Fan, Mini USB Fan, Mini Gooseneck Fan for laptop and PC - White: Electronics
Amazon.com: StarTech 5ft Desktop USB Extension Cable - A Male to A Female (USBEXTAA5DSK): Electronics
Amazon.com: Bluroo Universal USB to Wall Charger - Compatible with all USB Cables - 1-pack: Cell Phones & Accessories
Filters
As far as filtration goes, some of this you may already know. My golden rule to filtration is that there's no such thing as too much filtration. However, there is such a thing as too much flow. There are 4 main kinds of filters.
Hang-on-back, which goes over the side of the tank and.. Well.. Hangs on the back. It pulls water from the tank with a pump, flows it through a little box of filter media and then it waterfalls back over into the tank. These are your most common filters especially for tanks sub-55 gallons. They're easy, cheap, but they do have limited space and most usually can't hold much filter media, which is fine depending on how stocked your tank is, for instance one fully grown axolotl in a 40 gallon would be considered a very light "bioload" so it wouldn't need that much media.
Canister, which is a bucket-like container that sits on the floor below your tank. It will have a tube coming out of the top of it, as well as the side near the bottom. The top tube will go into your tank and pull water from it, where it will go into the top of the canister and flow through multiple layers of media, then it will be pushed out of the side tube which goes back into your aquarium. This is a great filter which are easy to set up and use, have lots of space and they are convenient. I personally like that the only thing that shows in the tank are two small tubes and you can put the canister in the tank stand so it looks pretty great. They're also pretty cheap, not as cheap as hang on backs but a lot cheaper than the next option.
Sumps, a sump is kind of complicated and kind of not. It's basically another fish tank, usually smaller, set up under your tank (in the stand again) and it will have glass walls arranged in a certain manner where water is basically forced to flow through a ton of media and then pumped back up to the tank. These are popular on huge tanks (75+) and are the best pure form of filtration. The only catch is that to properly run, without running the risk of overflowing your tank, you need to drill holes in your tank and plumb it so that when water is pumped from your sump to your tank, the water level will rise, and when it rises it will overflow into the plumbing and back down into the sump. This isn't really practical for smaller tanks but pretty much necessity for huge tanks. Another big plus is that it adds a ton of water volume to a tank, for instance if you have a 75 gallon tank with a 20 gallon sump, it's basically like having a 95 gallon tank, and thus you can stock it accordingly.
The last main form of filtration is sponge filters. These are very basic, easy, and cheap. All you need to power them is an aquarium air pump, which are cheap, and some airline tubing, which is also cheap. Basically the way it works you'll have a plastic tube with holes in it and a sponge wrapped around it, you put the airline into the middle of the plastic tube and the bubbles will pull water through the holes, thus through the sponge, and pull the water up the plastic tube and thus filter the water. Bacteria will collect on the sponge, that bacteria will filter your water.
A lot of people on here use a combination of hang on back and sponge filters. There are also a lot of diy filters you can make using bottles, pumps, media, etc. that you can find on youtube.
Media
Filter media can be a lot of things. Some are easier to deal with than others, some only work when used in a certain way. Easy and cheap things include aquarium sponge, 'ceramic media', nylon plastic dish scrubbers (non-used, no soap!), and lava rock. These simply require water flowing through them. Bacteria will form on them in time, this bacteria will eat the ammonia produced by the waste of your axolotl and turn it into nitrites, which is then eaten by your bacteria and turned into nitrates. Nitrates require water changes or a 'planted filter' to be taken out. A 'planted filter' is basically where you take certain kinds of house plant (most commonly the 'pothos' plant, cheap!) and put the roots in the filter. As water passes the roots, the roots eat the nitrates, and if you have enough of this you can potentially extremely long periods of time without needing to do water changes. More complicated media would be stuff like K1 Kaldness. K1 is used in what's called a "moving bed filter", basically you load an area in a sump with this K1, which is basically just little plastic rings, and then you put a powerful airline under them, this causes them to swirl around and crash into eachother, ensuring that only the strongest bacteria thrive on them and thus creating a no-maintenance form of filtration. Very handy if you can set it up, as the easy media I listed requires being rinsed every month or two. For small tanks you can actually take a plastic water bottle and load it with a liter of K1, drill some holes for water to come in and one small hole for an airline then just put it in a corner of your tank. There are a lot of videos for this online if you care to make one, a liter of K1 is like $7 on ebay.
Rocks
Rocks and gravel can cause big problems in an axolotl's life. Right now the rocks in the pictures are fine as the axolotl is just a baby still, but basically any rock that is small enough to fit in their mouth is unsafe. If it fits, they can eat it, if they eat it, they won't be able to digest it. If they can't digest it, they sadly perish. Like I said right now he can't eat those but definitely no rocks once you move on to a bigger tank. Some people use big river rocks (ones like twice the size of a full grown axolotl's mouth) as substrate, but I find that uneaten food, poop, and detritus (gross stuff that shows up in the low-current areas of a tank over time) will build up in and under the rocks, just a pain to keep clean. Most people use sand, but they say no sand until the axolotl is 4-5" long as even sand can cause digestive problems in very small axolotls. I myself prefer bare bottom but only with the bottom glass painted (on the outside of the tank).
I think that covers a lot of area, probably too much stuff really. If you have any questions feel free to ask. Sorry if I was a bit long-winded, but aquariums can be tricky especially when first starting out and from what you've already said you seem like you'll do great!