Hello! As you know I made a similar thing a while ago.
I used an ethernet shield. It polled a website which would then return a 1 to click one on or a 0 to stay the same. It'll be easier for you as you're displaying a number.
You need a counter on your website. Set up a very simple page that shows the number - you don't need to bother making it a html page, just show the count.
During development you'll want to see what you get from the web server and work out how much text you need to discard before you get to the number. For ease you may want to pack the number with zeros so it's a fixed length - remember it'll be easier to get your website to do more than it will to get the arduino to do it. So, figure out what you don't want, read and discard that number of characters then store your the numbers into one or more variables.
Now, as for the nixies - I've made a nixie clock. I've heard rumours that displaying the same number for a prolonged period will damage the tube, so I have my clock randomly flicker through random numbers for a while at a random interval. You might want to incorporate some change, maybe scroll through them when there's a change?
Sadly I've not done anything with my arduinos in a while. I got married recently and am now redecorating my entire house so all my stuff is in storage, including the PC I use for development, so I can't give you any code. Let me know if you need any help though, and please message me when it's completed, I'd love to see the final thing!
edit: As for materials, any arduino board should be fine. I'm still on duemilanoves. You'll want an ethernet shield (get one which handles the tcp/ip stack itself) and whatever display you decide to use.
A nixie display would be cool, but you'll also need a HV power supply (you can get them on ebay) and your chances of procreating something up increase as you mix put the high voltage up the wrong bit (I did and killed an arduino).
You can pick up LCD displays nice and cheap and they're easy to set up (see http://learn.adafruit.com/character-lcds/character-vs-graphical-lcds
I've got a few Sure Electronics 32x8 LED displays which are straight forward to use, but not as simple as an LCD display.
I suggest you get a cheap small LCD display and get the thing working, then if you want you can change the display.