The objective of 1 carryon plus one personal item pretty much follows common sense.

Your carryon should be able to fit in the overhead bin. Your personal item should be able to fit underneath the seat in front of you.

You might check the UA's British site. The requirements coming home may / probably is different than going to.
http://www.direct.gov.uk/en/TravelAndTransport/Publictransport/AirtravelintheUK/DG_078179

One thing to remember, is that you're going to have to haul that stuff around.

Don't forget to read the TSA 3-1-1 on carryon liquids.
http://www.tsa.gov/311/

Your free check bag is probably 50lb limit (22 kilos or similar). You'll pay extra for overweight.

Overhead bin sizes vary on different types of aircraft. Some overhead bins are really big, and your bag should fit just fine. The flight attendants should help you find overhead space, or allow you to check your rollaboard, should there not be space for it. You have no status with the airline so will probably board in one of the last (if not THE last) group, which means space will be at a premium by that time. If your bag does not fit, the flight attendant (or the agent at the gate, before you board) will let you know that bins are full and you will gate-check your rollaboard bag, at no cost. Your computer bag will remain with you, and yes, it should fit under the seat in front of you.

You will see plenty of people who violate these rules! They bring everything but the kitchen sink on board, and their 2nd bag is usually a pretty hefty sized duffle, which they throw in the overhead bin along with their rollaboard, which leaves less room for people who do follow the carry-on rules!

Enjoy your flight! I hope your studies go well.

I'm a frequent flyer on UA and I have never seen them measure a carry-on, and it is as plain as the nose on your face that many people's are too large.

I travel with my wife and we recently bought a very large stroller (a four wheeled bag) that was just under the maximum dimensions allowed by UA, which we check. It is huge and my only concern is that if I really packed it, it could be over the weight limit imposed by UA. Other than picking it up when it gets full of wine bottles, it is easier to manuever than a roller bag. Look for one with handles on every side (including the bottom) so it is easier to pick up. I also put a nylon belt around it to keep it from opening and make it easier to see on the carousel.

The reality of carry on luggage is that very many people take on more than they should, and the overhead bins are smaller than they should be, and the airlines have hidden a lot of equipment under the seats. So it is almost a certainty that, unless you are among the very first to board, and UA now assigns you to a group so its just luck if you go first, they will run out of space and gate check some carryons. That means they put a tag on it at the gate, give you a receipt, and put it in the luggage hold. The good side of this is that they do not charge a fee; the bad side is that if you have anything valuable in it, it may be lost. I have not had to gate check anything in years, but when I did they brought it up to the plane door on arrival. I suspect they may no longer do that, since there would be a pretty big pile there.

Assuming you will want your computer case at school, I would pack it, empty, in your checked luggage. Get a back pack big enough to hold your computer, camera, medicines, etc., and use that as your personal item; they are usually malleable enough to fit under the seat in front of you (don't get a bulkhead seat, they are narrower and you have no underseat storage), where you can keep an eye on it, easily get what you want out of it, and use it as a footrest. Don't let them gate check the knapsack; tell them it has stuff in it you will need during the flight.

Since you will be there a while, you might look into mailing or shipping some of your things. It is not terribly expensive if you don't need one-day service. I'm sure the school has an address you can use for that.

Thanks for the advice, guys! I just wanted to make sure I was following the rules and wouldn't end up getting penalized for my bag sizes. I've heard of bags getting gate checked, but does that happen exceedingly often?

I was planning to put everything essential in my computer bag (electronics, prescription medication, etc.), so as to make sure I would have those things on me at all times. So even if they need to gate check one of my bags, I can still keep my computer bag with me, yeah?

Ok so the reality is, if you are going in July and are on a packed flight, everyone will have the same idea as you, pack as much as you can into a carryon and see if you can bring it on board. As Simpson has said, without status, you will be the last to board so there may be no be room in the overheads at all. And gatecheck happens on full flights in the summer....
So it's really NOT so much question whether your carryon fits the specs as much as there likely won;t be room for it and you will HAVE to gate check it. Get a small padded computer backpack for your computer and essentials and make sure it stays with you as your personal item.

If you have a standard size computer bag, it should easily fit underneath the seat in front of you. Your roll-aboard may well need to be gate-checked, as overhead bin space is never enough for all the stuff people bring on board. I assume you'll also be checking a bag.

My experience is that on international flights, they are more vigilant about what people bring onboard. I have seen them walk through and take bags from people who have more than one carry-on and one personal item and check them at the gate. So, if you are lucky, they may enforce the carry-on rules and you will have overhead bin space to stow your carry-on.

Well, now that I've been thinking about it, I might be able to squish everything into one checked bag if it's big enough, thus eliminating the carry on bag issue entirely. All the essentials can fit into my computer bag (medicine, electronics, a change of clothes in case, etc.) We'll see, though. It depends on how big my checked bag ends up being (because I don't have one yet).

Although, if I do eliminate the big carry on bag, could I squeeze by something else with my computer bag, like, say, a camera bag? I have one of those nice DSLR cameras that I wanted to take along (not only for personal pictures but also some for the study abroad center at my college, so they need to be high quality), but I thought I would need to cut because it wouldn't be able to fit in my computer bag. If I lose the big carry on, would they let me by with a standard computer bag and a small DSLR camera case?

Take a backpack AND a laptop bag. A lot easier to find space in the overhead bin for a backpack than a rollaboard. And worst case, both can fit under the seat in front of you. That's how I carry my dSLR and lenses and laptop on board.

Yes, you can carry on both a laptop bag and a camera bag.

simpsonc510 - Check out this recent New Yorker cover. Sounds exactly what you were talking about.

http://www.newyorker.com/online/blogs/culture/2012/04/cover-story-travel.html

Hey, rastaguy, I was on that flight!!! lol But it does have a message.

Are you going from home on a nonstop transatlantic flight, or will you begin with a domestic flight to connect to the international flight? If all your travel is on the big transatlantic plane, you probably have less to be concerned about than if you will connect from one plane to another. I say this because the past few years nearly every domestic flight I've been on has been full, with many passengers bringing roll-aboards to avoid the $25 bag fee, and a good many of those getting gate-checked. My observation on transatlantic flights has been, generally, that the overhead bins are roomier and, with one free checked bag, relatively fewer people try to force large bags into the bins.

A soft duffel or backpack that will hold your electronics could be either a "carry on" that can squeeze into the bin, or your "personal item" that will go under the seat.
Safe travels.

@kayd- I'm flying transatlantic straight out of Dulles to Heathrow and back again, so I don't guess domestic-only problems are on my radar. =)

Do not put your camera in anything that might possibly get checked or gate checked. Far too much chance of theft...and it happened to us. Nothing valuable in checked luggage!