Ölgii is the capital of Bayan-Ölgii province, and is also known as Ulgy or Ulgii. Nestled in the Altai mountains in western Mongolia, most people in the region are part of the Kazakh ethnic minority, with a small percentage of Mongols living in the province.
Routes to enter include coming by plane to the Ölgii airport, usually from either Ulaanbaatar in Mongolia or from Kazakhstan; routes can also include biking in, driving in by bus (from Ulaanbaatar or the Altai region of Russia), or by private taxi/transit.
Ölgii is not a large town so you should be able to get most places by walking, 30-40 minutes at the most. Unofficial taxis are cheap and can be found alongside major roads or in/near the city square. You can signal for a taxi by sticking your hand out waste high and flicking your wrist. As of 2013 it costs around 600-1800 togrog, depending where you're going, more if you don't speak Kazakh (The exchange rate as of 1/25/2013 is 1395 togrog to $1). Most drivers will speak Kazakh, but can speak Mongolian; bring a pencil and paper to write down numbers (in Tg) if you don't speak either.
By going to the bazaar you can also hire a shared taxi which is going further away into nearby towns; Only Mondays baazar does not work. Most taxis leave around 3-4 pm afternoon.
Check out the local museum with displays of Kazakh culture.
Green Garden is a nice place to hang out in the evenings; has cheap beer and soft drinks, and is one of the greener spots in town. Most tourists come in the summer when it is quite brown and dusty (in the winter it's snowy!), so green is a rare thing.
It's worth going on hikes up into the nearby mountains; find someone familiar who can guide you there.
It's also worth experiencing Kazakh hospitality in a ger, or white felt round-house. Many people set up gers in their yards in the summer, but you can also go outside of town to see gers in their "natural setting."
If you happen to be in town during Naadam (the Mongolian summer festival, usually in early July) there's a nice little celebration outside of town up on the hill, with horse racing, drinking, and lots of people picnicking and shopping at tents set up there.
Visit during late September and early October during the two eagle festivals. The first, the Altai Kazakh Eagle Festival in the nearby town of Sagsai is the last weekend of September. Then, the Golden Eagle Festival just outside of Ölgii is billed as the largest gathering of eagle hunters (hunt with eagles) in the world with 70 participating in the event. Both events feature skills competition among eagle hunters and traditional Kazakh horse games.
Most people come to see the surrounding areas and don't spend a lot of time in Ölgii itself. It really helps to know someone local who can introduce you to other people! Learning at least a few words of Kazakh will also be helpful and people will really appreciate it.
Altai Craft is a providing women with employment by sewing. You can look it up online. They have some beautiful Kazakh embroidered bags, pillowcases, placemats, coats, hats, and even Christmas stockings! There's also an Artshop on south east corner of main square, closer to the center of town, which also sells tourist handicrafts, Kazakh hats, and clothing.
Kazakh Craft is an Ölgii based maker of traditional Kazakh embroidery and eagle hunter accessories. Their main shop is on the main road. Though most of the actual embroidery is done at one of the 40 women sewers' homes. During the summer they have a shop in a trailer in front of the museum. Their products can be found in Ulaanbaatar and Kazakhstan. 
There's a small bazaar, mostly filled with cheap imported stuff from China, but also some local stuff.
There are lots of small shops in the center of town that also sell food and odds and ends.
Manti (dumplings) is a must! There's a shop just off the center of town, facing the "BU Palace," which makes delicious manti soup. Has some other reasonable dishes as well, and usually has beer and soft drinks.
Several smaller shops will have meat patties or manti for cheap, usually served only with milky tea.
If you're lucky, you'll find a shop or friends that serve delicious bisbarmak (Kazakh traditional food - lamb and noodles) or kuz (fat mixed with dark lean meat - very tasty!)
You'll probably find it difficult to eat well as a vegetarian, as Mongolians and Kazakhs rarely eat meals without meat. Though, the bazaar and some supermarkets have decent selection of fruits and vegetables. There is a regular supply of bananas, apples, pineapples, carrots, bell peppers, cabbage, oranges, and watermelon throughout the summer and fall.
You can find Coke, Sprite, and Fanta in most shops; Diet Coke is found in fewer shops.
Several stores will serve beer and you can also buy alcohol at some of the larger supermarkets, although you may find it difficult to buy alcohol on Fridays because of religious observance. Most Kazakhs will drink alcohol, even though they will say that it isn't appropriate according to Islam, but they may not drink on Fridays.
Check out the Duman hotel for moderately priced rooms ($20 or so a night).
For a longer stay, it's possible to rent a local apartment for around $150-200 a month, if you can find someone to translate and advocate for you.
Nearby destinations include several-day excursions into other areas of Bayan-Ölgii province to see the beautiful mountain scenery, Kazakh herders, and ancient petroglyphs.