So everything DVM and orca have said is on point. I was also checked for proof of attending university the first time I entered, so make sure you bring those letters. And definitely double check voltages on electronics - I brought my chi straightener and a nice hair dryer only to find out the voltages wouldn't work, so I had to bring them home later and bought new ones here.
Outside of that, I mainly only brought my electronic chargers, clothes/footwear, and a few sentimental items. Clothing is pretty expensive over here, so I really only do my shopping when I'm on visits home in the US. Any apartment stuff you can purchase for fairly cheap at an Argos or elsewhere, so don't waste any suitcase space on that sort of thing. Forget about books and folders as well, like orca said they use A4 here so it's too large for anything you'll buy at home. A rainjacket is a good idea, I had an umbrella for about a week before I stopped using it, it was usually too windy and the rain was never really more than misting/light that a rainjacket worked for, if it did downpour it would stop fairly quickly so I'd just wait it out. Oh and in regards to any medication, you do get registered with a GP so most stuff you can get here for free. I'm not sure about any ongoing prescriptions you might have, but I was on birth control that I brought with me, and after I ran out I started getting it from a gyno here instead since my health provider at home wouldn't mail medications internationally.
In regards to money - check to find out what your best exchange rate option is. It's criminally high in a lot of places (I can't really remember what I did, but I vaguely recall bringing a travelers check thinking it would be fine, but later discovered had stupid rates), and in the end it worked out that the cheapest option for me was actually through my bank at home. I usually would just transfer directly to my UK account or if I didn't plan ahead and needed to withdraw a chunk of money from my US account at an ATM here, the transfer rate would be decent.