UPDATE: Moving this to a page of its own. Any further updates on things to look out for will be done there.
Especially important for Erasmus Mundus students because travelling means the costs pile up:
Visa fees: Even though the EU Visa Code says students don't need to pay this, some consulates and embassies will lie to you.
Residence permit: Apart from the visa, you'll need a residence permit for most places. Free in some places, painfully exorbitant in others.
Import Tax and VAT: Alex Berger has a nice post here detailing the absurdity of the Danish postal service where fees and taxes chalked up to 130% the cost of the actual object. And, "Denmark charges taxes on fees and taxes". This is particularly important because, while my experience is limited to Austria and Aalborg at the moment, Europe may not have a good electronics ecosystem. I wanted to get several items including a camcorder and a pair of earphones. All of these were not available both in physical and online stores based in Europe. Importing would mean too much in shipping and taxes. Furthermore, if you are likely to work with samples, then you will have a lot of fees and taxes to pay in some European countries. This is definitely something to check out. It can put serious dent in research. I've more or less decided not to build my career in Denmark based on this criteria alone.
Air, train, coach and metro fares: Be sure to note the websites you can use to get tickets. Carpooling is one good way across Europe.
Luggage: A lot of old buildings in Europe do not have lifts/elevators. Chances are, you will find one with a lift to live in. But if you're travelling, chances are, you'll meet a lot of these liftless structures. So try to get a bag with both wheels and backstraps. I don't have a solution for that, but I like the well-built Tom Bihn Aeronaut 45. I first found it on Wirecutter, a really good website that does well in its market niche.
Laundry: You might want to get the Flexo-Line. We had so many issues with trying to find a place to dry our things. Our dryer kept breaking down, for one thing. It'll come in handy if you're travelling as well.
Bank fees: Some banks charge you fees, some don't. Some banks in Austria think if you're above 26 you're too old to be a student.
Change money: Do not do it at an airport.
Rent and accommodation: Be sure to check with previous batches if your potential accommodations are run by villainous landlords or if there are scams. We had some really bad and disappointing experiences, even with institutional providers like universities. You'd think they would be professional about it but, no.
Living costs: I'm still trying to figure out how best to represent this. Maybe by using purchasing power parity (PPP)? I don't know. I'm not an economist. Most people aren't either. But obviously, there is a difference between living in Poland and in Denmark. You can check out my expenditure as I document it here. Thrift stores are always a great idea. There are good online deals too.
Online shopping: Chances are, you will have trouble with this. Sometimes it is because you have to open a bank account in Europe for scholarship funds (some programmes will transfer money only to banks based in the country from which they are administrated), convenience and other reasons. Your card won't work. Amazon won't trust you until you fax or snail mail them your bank details and one or two other documents. You can either verify yourself or keep your bank account and card from your country of origin, which also means you need to keep your phone number from home, at least till you get a new one and update it with your bank back home.
Phone plans: To save money, do not get mobile data. In Austria you can spend 10 Euro (at A1, I think) and make more than enough phone calls and SMSes for months if you use it only to locate your friends while out together or to call about official things now and then. In contrast, mobile data comes at roughly 10-20 Euro per month, cheapest.
Email, Cloud Storage, Double-Authentication: I cleared my browser cache when I came to Austria. When I tried to log into Gmail and Dropbox again both wanted me to authenticate myself using my phone. Unfortunately, I'd terminated my phone number in Singapore. It took me about 4 days to deal with Dropbox. It took about a month to convince Google I was myself and it was aggravating. So, turn off double-authentication, use another email address for it, or keep your home country phone number.
Computing: If you don't already have a computer, get one in your home country before coming over. The keyboards here will be different based on the native languages. You won't be able to access things on your computer easily and you will become super annoyed.
Power point adapters: Awfully useful. I suggest getting at least two.
Rice cooker: If you're Asian, you might want to consider this. Difficult to find a small one here. I do not recommend this because I prefer going native. But hey, no judging.
Winter wear: Different regions require different layers. If you have never experienced winter before, please go shopping with people who know what to get. Otherwise, you will buy some really useless items. You will. We all make stupid assumptions when we do not know what to ask about what we don't know. Experience is the best teacher here.
Very important: Always consult those in previous batches of your programme before you come over. Do not be afraid to ask stupid questions. There is no way you can know if certain assumptions, even about the most obvious things, are totally false.
Did I miss out something? Let me know. I'll continue to update this.