Or can I just walk into my local bank?
Yes, you would usually approach your bank. However, you wouldn't just "walk in", you would ask for an appointment with a financial advisor.
Will a bank take me (a 20 year old university student with 2000 euro in excess savings) seriously?
Of course they will. Most people pick their bank at your age and then stick with it for the rest of their life. If they advise you well, then there is a good chance that in 5 years from now you will open a retirement savings account with them and in 10 years from now you will ask them for a real estate loan. So banks are very eager to acquire new customers in your age bracket.
Just make sure they don't try to sell you things you don't actually need. If the banker happens to be a good salesperson and you say yes to everything, you might walk out with a couple insurances and saving contracts and student loans you didn't actually want or need. Don't sign anything yet. Ask to take the information material home with you. Study it, do some Internet research to find out what they are actually offering you and if it's a good deal compared to what other banks offer, talk about it with people around you and ask them what they think about it. If you are totally confused about the implications of a financial product you are being offered, feel free to ask another question here.
Then, when you are sure you know what you are signing and that it is the right thing for you, make another appointment to sign.