Erzurum is the highest major city (population: 360,000 according to 2000 census) in Turkey, situated at an altitude of about 2000 metres over the sea level. Complemented with the distance to the sea, this makes the climate of this area the harshest in Turkey with the temperatures in long, heavily snowy, and bitterly cold winter regularly below -30°C at nights (and no warmer than -15°C during the day). Wet summers in Erzurum are around 30°C in the day time and is chilly at night time, although temperatures dropping down to freezing point during summer are becoming rarer and rarer each year, probably with the advance of the global warming.
The city hosted the international 2011 Winter Universiade (“university olympic games”) .
There is a daily train from Istanbul (Doğu Express, departs from Haydarpaşa station at 08:35 every morning). Doğu Express has a stop in Ankara, too (and in a number of other cities and towns across Anatolia, including Eskişehir, Kayseri, Sivas, and Erzincan among others as well). There is also another daily train from Ankara (Erzurum Express, which is said to be more comfortable than Doğu Express). Both trains head for Kars — further east (near Armenian and Georgian borders). See Turkish Republic State Railways website for schedules and fares: 
Most bus companies stop at the Erzurum Otogar, which is about 1 km away from the main city centre. It is a walkable distance to town, or you can take the public bus to/from the Otogar.
Buses from Trabzon are available (4.5hrs and 30LT, 25 if you don’t look like a tourist and don’t speak much english).
Erzurum is a pretty small city, and you can cover most of it by foot, unless you are thinking of heading out and around the city, in which case, most public buses and taxis will be able to get you there.
Citadel of Erzurum - The Citadel of Erzurum is situated on the top of the hill in the middle of the city centre. The YTL 3 admission fee allows you to explore the grounds as well as to climb up the minaret/clock tower, as well as visit the citadel masjid. It's worth the effort to climb up the clock tower as it gives you a commanding view of the city as well as the surrounding mountains.
Ski in Palandöken ski resort, which is only 10 minutes away from city centre by public bus.
In summer you can hike up to the top of Palandöken (3,150 m.a.s.l.) for fantastic views. There is a road (first paved up to some of the hotels, then a dirt road) up to the very top, but it is possible to take shortcuts when hiking. There are no marked trails, but the directions are clearly visible. It is good to start early to allow time to enjoy the views.
Oltutaşı (Oltu stone in Turkish) — This is a kind of amber which is extracted only in Oltu, a town near Erzurum. It is mostly chipped into jewelry, with different levels of quality and correspondingly different levels of prices. Look around in Erzurum. Taşhan is the best place to shop for Oltutaşı and it is also an interesting traditional building.
Silver — In jewelry shops in the center or on Taşmağazalar street, fancy authentic silver jewelry can be found.
Stockbreeding is widespread in the region, and there is little chance to cultivate vegetables and fruits in this high and cold land. Therefore traditional food is mainly dependent on meat.
The city is also known for different types of local cheese that is hard to find elsewhere in Turkey.
Kadayıf dolması — A must-taste local dessert of Erzurum cuisine. It is made from kadayıf, a sweet pastry that is lots of thin fibres of flour put together. Walnuts are stuffed in the kadayıf and the stuffed kadayıf is first fried, then cooled and finally ready to serve with addition of a sweet sauce. It may be sampled in virtually every restaurant in Erzurum, but some of the best can be tried in a specialist restaurant, Muammer Usta
Döner — Typical Turkish döner is really made well and should be tasted in Hacı Baba Restaurant on Emniyet Caddesi.
Cağ kebap — local kebab of Erzurum, that you can have in Gelgor Restaurant. It is made with lamb meat and the preperation is like döner, but unlike döner, instead of a vertical axis, the meat is cooked on a horizontal axis.
Güzelyurt Restaurant — A very nice old restaurant right in the city center on Cumhuriyet Caddesi. You can have a good Turkish meal with some Turkish raki and enjoy the atmosphere. 
Hmmburger — A very tiny small hamburger place. Hamburgers are really delicious.
Coffeebox — A local coffeeshop, with various pasta options and nice coffees.
Renaissance Polat Hotel. Four stars hotel. 220 rooms. TV satellite. +90 442 232 00 10
Yeni Cinar Hotel, (close to Istasyon Caddesi). Fantastic, sparkling rooms for this price. Complete with satellite TV (only Al-Jazeera in English), large bathroom and free WiFi.30 TL per night for a single room (huge, double bed) in May 2010. Easy to negotiate down to 25 TL per night if you are going to stay two nights. 60 TL for a double room (November 2012). (39.9105,41.2755)
Cinar Hotel, (Right next to the Yeni Cinar Hotel). Rooms seemed a bit more cosy than in the Yeni Cinar Hotel. Private bathroom and free WiFi.40 TL per night for a double room in November 2012..
Bahar Hotel, (In front of the Cinar Hotels). Small room without private bathroom. Free WiFi.30 TL per night for a double room in November 2012.
ARI otel. On the road to the Yeni Cinar Otel. Very helpful and friendly non-english speaking staff. 60 TL (bargainable to 45 TL) double room with shower in the room but toilet at the floor. Aug 2012..
The area code of Erzurum is 442. Dial +90 442 when calling from out of Turkey.
Dogubeyazit, Kars, Sarıkamış, Van, and Iran are nearby and further eastward destinations. Buses run by Star Ağrı Doğubeyazıt heading for Dogubeyazit (25 TL, 4 hrs) depart daily at noon, 3PM, and 5PM on weekdays and at noon, and 3PM in weekends.
If the Seljuk sights in Erzurum did not satisfy you sufficiently, you may try Divrigi with its UNESCO World Heritage site on your way to west. Two daily trains from Erzurum call at town's station.
Buses to Diyarbaikir take 6 and some odd hrs (from 50TL tourist price you can bargain down to 35TL)