Living in Leipzig
The University will offer you a room in a hall of residence (Wohnheim) which you can accept or reject up to two weeks after moving in. In practice many exchange students find they would prefer to find a shared flat (WG) in order to have more chance of speaking German. As this might well be true for you, be prepared to do something about it within those first two weeks. You can look for a room in a WG either on the noticeboards at the uni or at Das Schwarze Brett im Internet. Many potential flatmates will wish to meet you for a chat before offering the room and before you can decide you want it. If you are looking for a sociable, friendly shared flat, avoid those which advertise "Zweck-WG", which is the opposite (i.e. people sharing a flat for only practical or economic reasons rather than in order to enjoy each other's company).
When you move out you will be expected to re-paint the room white (unless it is completely spotless), or you will be charged a fee from your deposit.
Rooms in flats are usually offered completely unfurnished. You may be able to get a furnished room by taking a "Zwischenmiete" - temporary renting from someone who is away for a semester or a year. As with all shared flats this can often work out really well - but that cannot be guaranteed in every case. Among other highly useful pages, Haus der 5 Kontinente has a good description of the different areas of town you may find flat rooms offered in.
The university lists here some of the places you can stay temporarily until you have found somewhere more permanent to live.
Deutsche Telekom (T-Com) will connect your phone line (eg in your Wohnheim room) for 25 euro, if the last disconnection was within 3 months. Otherwise the fee is 50 euro.
Once connected, you will pay your line rental to T-Com but you can charge your calls to a wide range of other companies at much lower cost, simply by dialling a 5 or 6 digit number (eg 01030) before the number you are dialling. In this way it is possible to dial internationally as well for next to nothing. For the best codes to dial at certain times of the day to specific destinations, check out billigertelefonieren.de - and note that the codes and tarifs change frequently so update your list regularly. Note that in most cases, even though you pay the lower tarif of the alternative companies, the amount is billed via your T-Com bill and your payment is made via T-Com, which means you do not need to register with the alternative companies - just dial the number.
An alternative to getting an actual landline is to get a homezone on your mobile. This was pioneered by O2 with its Genion tarif. As well as a standard mobile phone number, you are issued with a local (landline) number which works within a 500m radius of an address which you specify. When you are in that zone, people can call you on your local number, and you can make calls at cheap rates (eg 3cent/min). When you are outside that zone, people dialling the local number are diverted to your voicemail, and your outgoing calls are more expensive. Of course, people can dial your standard mobile number wherever you are (inside or outside the homezone) and they will be connected to you. Vodafone and T-Mobile now offer similar tarifs. Note that you cannot take advantage of the low-cost prefix codes (see previous paragraph) from a homezone mobile. You should also check the minimum contract lengths of homezone tarifs.
If you want an ordinary mobile deal, the above operators, along with eplus, have the major networks. Pre-pay offers are numerous and constantly changing, so you'll have to hunt around for the best deal (teltarif.de may be able to help). But for starters, try blau.de.
Internet High-speed internet is included in most Wohnheim accommodation. In shared flats, it is often offered as part of the package (most commonly known as 'DSL', or 'DSL flat' to indicate an unlimited broadband deal). Otherwise you may have to arrange it yourself, with main operators being GMX, freenet, 1und1 and arcor. However, do be aware that the minimum contract is often 24 months long.