1 Week In Romania

What you should visit depends on what your interests are,what things do you like to do?
Your costs are going to depend on how long you stay in one place and how much travelling you do.With only these bare details you give,maybe 30-40 Euro per day max.

Basov is the main place to visit in Romania, and the various castles nearby, theres good hiking too if you like that. Bucharest isnt worth longer than 2 nights, or even 1. I'd also highly recommend Suceava in the north-east, the painted monasteries near there are beautiful. Sibiu and Sighisoara are nice historic towns worth a night or two as well. For budget travel, plan on 25-35 euros a day for everything and thats living quite well, Romania is a cheap destination.

I spent 8 nights in Romania 2 years ago. This is what I did.

3 nights Bucharest (2 full days).
3 nights Brasov (2 full days)
1 night in a small village in the mountains (can't remember the name as I was taken there by the Romanians I was Couchsurfing with)
1 night Sinaia.

Sinaia could be skipped, I went there because I hardly ever get to see snow as where I live in Australia it never snows.

There are hostels in many built up areas of Romania like many countries some are a lot better than others. If you want a rural hostel there is a very nice one in the traditional village of Breb in Maramures. Food can be brought in most markets and there are cheap restaurants around but there are also a lot of pretentious restaurants and hotels that will eat your wallet so be careful. Trains are a good cheap method of transport can be a bit slow but obviously you can save on accommodation by using over night trains such as the one between Bucharest and Sighet in Maramures which has couchettes. 35 Euros yes just about if your careful if wont give you much room for the simple luxuries in life.

Yes for Sighisoara, a sure nice small town.

For an easy going historic city, with a large student population and some night life, an even bigger yes yes for Sibiu.

Kindly note:
because of family connections my views are influenced.

Hotel Trianon 3* could be a good place to stay right in the old city center of Bucharest closed to Cișmigiu Gardens. Spacious rooms, Good restaurant, Wireless connection available! Good value for money!

Itinerary ideas: http://www.romaniatourism.com/sample-itineraries.html#1

A room in a three-star hotel will be about $70 + -,
but you can also rent a studio in an apartment building for less.

$20 / day will be more than enough for food (even less).

Public transportation is quite inexpensive

Skip Bucharest. Spend time in Brasov and Sibiu. If you can make it to the north the area of Maramures is an absolute must. The Brasov/Sibiu area feel very much like western Europe. Maramures, on the other hand, leaves you feeling like you've traveled back in time 50 years. Much more unique in my opinion.

One week isn't enough to visit all Romania, so it depends on which are your thoughts ! What r' u planning to visit, which part of Romania and what budget ? Where are u flying ( or travelling ) to Romania ? What exactly are your interests ? Glad to help you if u give more details !