Julie,
In my experience, Hungary has everything you might need. Just bring clothes that can go through the seasons because they have all seasons here, and be sure to have some shoes that are water proof/resistant for the winter. I would say that good quality well made clothes are more expensive here than they are in LA.
You can easily buy a car here so there is not a need to ship one unless you live very close. It is very easy to get around the city in Budapest because public transportation is very good.
It is important to note that many of the freeways you have to pay in advance to drive on them or you can get a very pricey ticket. It is a little difficult to tell when the toll starts and when it stops so I would suggest that if you plan to travel alot throughout the country by car to just buy a pass which you can buy in almost any gas station. It is not very expensive for a month but you can also buy it by day and week.
And also, to make sure you have a good map that even shows the little villages/towns because, unlike the states, the road direction is determined by the next town or city and not by the actaul direction that you are going (south, east, etc.) So if you don't know the towns that are on the way to wherever you are going, it is very easy to get lost.
In Hungary, in my experience, they have all the modern conviences as far as appliances, blow dryers, bathroom products, etc. and the markets have a good array of produce and foods.
There are also health food stores and in regular stores they sell gluten free products if you might need those.
For me it was a very easy transition, just have a good Hungarian to English dictionary so you find specific shops and ask for certain items that you may not recognize because it is a different brand.
In Budapest you will find a lot of English speakers but in the surrounding cities, not so much.
The one thing that they do not have is dryers, so be prepared to hang your clothes to dry and take into account that it takes a day or two for things to dry so I would recommend bringing multiples of anything that you even think you might need! When it rains and you get soaked, its not like you can throw something in the dryer real quick. Of course you can buy clothes here as well.
It took a bit of time for us to find the right flat to rent out but there really is a nice selection of places for rent so you shouldnt have any trouble finding a place.
I just took for granted that everywhere would have air conditioning but that is not the case and it can get very hot and humid in the summer. Especially if you are on a top level floor.
My advice is to ask for air conditioning and make sure there are plenty of radiators in your apartment for heating in the winter. We once stayed in a place that had only two heaters and they were old so the metal would contract in the middle of the night which would wake us up everytime. Most places have decent heating - we just didn't know because where I am from it's forced air heating and no one has radiators. So be sure to ask!
The other thing is that at some point most buildings have to turn on the heat for the building when it reaches a certain temperature outside. Be sure to ask if there are any limitations because I have a friend who lives in a building and they go by a certain date and not temperature so she was very cold until they turned the heat on and things are not regulated well here so she really just had to suffer.
I hope this was helpful and if I think of anything else I will be sure to post it here
