Mongolia is a treasure trove of mountain sport opportunities. While none of the mountains in the country are superlative in any specific comparison to other great ranges of the world, the remoteness and lack of development outside of the capital makes this one place where you can truly feel like you are exploring something untouched.
I would break up the mountain sport oppotunities in Mongolia into two categories; 'near Ulaanbaatar (UB) and 'everywhere else'. In short, if you think you are going to get beyond the small bubble accessbile within an hour or two from UB, it requires somewhat of an expeditionary mindset.
Around the capitol are 3 places you can go that provide some level of mountain sports, all of which have develop support infrastructure (lodging, dining, electricity, etc)
Terelj National Park
This is the big one, and deservedly. This national park is a beautiful cluster of rugged peaks that offer some genuine, albeit sketchy rock climbing opportunities. There is even an absolutely gorgeous luxury hotel and resort. While I know the audience here is more of an outdoorsy crowd, I highly recommend a night or two at the hotel. Oddly enough, I ended up staying there when mid-winter, my land cruiser broke through the ice crossing a nearby river (I escaped out the window just fine lol) and I needed to wait a day for someone to bring out a replacement vehicle for me. It is a beautilful hotel, plenty of well-made traditional dining options, and a pretty nice spa/pool.
This is a small resort area (the term resort is used quite loosely here) 2-3 hours northwest of UB. It is surrounded by a number of crags with some climbing potential, along with a large monolith that has developed climbing routes. There are a few ger (Monoglian word for yurt) camps and one wood log lodge. The lodge is really cool, not expensive, and I would strongly recommend staying there! Aside from the climbing, there are scrambling and hiking opportunities as well.
This is the mountain just south of Ulaanbaatar, the only mountainous area accessible directly from the city. It has a few small crags scattered around it that have been sort-of developed, or at least identified. Mostly, this is a hiking and cross country skiing destination. Near the northeastern edge you can find Mongolia's only lift accessed ski resort. One of the hiking trails here is described on hiking project, and some of the climbing areas are listed on mountain project.
This list could go on forever!
Ihk Gazaryn Chuluu: A cluster of desert columns and rock stacks halfway between UB and the Gobi. From their look, definitely climbable, but little to no real information about it. If nothing else a cool spot to camp overnight on a road trip.
Gobi Gurvan Saikhan National Park: West of the desert town of Dalanzadgad, a range of craggy rock that does look a little crumbly but could probably make for some interesting scrambling. The main attraction here is the year-roud ice deep in the gorges of eagle valley, and the river which freezes over entirely in winter.
Khovsgol Lake Area: This lake is a huge attraction, especially for Mongolian Holidays. To the west of the lake is a mountain range with very minial climbing information, however a few youtube videos of people climbing, and map analysis indicate that this would be an area with potential for spring alpine-esque climbing or summer/fall scrambling. Rock looks somewhat chossy, normal for this climate, but overall lots of potential.
3-4 Day Drive
Otgon Tenger Uul: This mountain has its own page here already here.
The Southern Altay Mountains: This is the arm of the Altay monutain range that exends west to east along the southern regions of Mongolia. There are numerous higher altitude paks, some with glaciers, all along this range. Most have no climbing information and would be quite an adventure.
4+ Day Drive (plus horseback or foot travel)
Now we are getting into some of the real expeditionary style trips in Mongolia.
Khovd Province Area: In this northwest area of Mongolia there are plenty of mountains waiting to be climbed. One area, Turgen National Park, has a couple peaks with minimal but still existant information. I found one youtube video.
Altay Tavn Bodg National Park: This is a huge national park with innumerable glaciated alpine peaks waiting to be climbed. The main attraction is Mt. Khutien, the highest peak in Mongolia and a peak that forms an interesting tri-border with Mongolia, China, and Russia. This peak has quite a bit of information for climbing since it is the most popular alpine climb in the country. Take some time to see the rest of the park, there are many UNESCO heritage sites throughout. Some very old history of human migration is preserved in this park.
Mongolia is a huge country, there is undoubtedly more to find than listed here.
Flying into UB is really the only option, from there you may find local flights to other airstrips around the country. However, the best part of Mongolia is the unlimited overland expeidition opportunities. I would absolutely suggest that you rent a 4WD vehicle for self drive. The driver/guide-provided rentals are cheaper and receive great reviews, but what is the fun in having someone who already knows all the answers??? You may as well go to Disneyland at that point. Go it alone and forge your own adventure. I always used Drive Mongolia. You can get maps with general-locations of gas stations mapped out, which takes care of the biggest risk, which is being stuck in in the middle of nowhere with zero communications. If you can read a map and navigate, you'll be fine. The gas station attendants in the rural areas may very well try and give you "foreigner prices" if you don't have a guide, so either learn enough Mongolia to stick up for yourself, or just bring a little extra cash/goodies to give them (tobacco products worked for me).
Climate will have a massive impact on your Mongolia experience. In a nutshell; dry and cold, but can also be balzing hot and dry specifically in the Gobi.
Spring (April-May): Temps will range from below freezing to 60-70 F. Fairly comfortable time to be there, occasional precipitation, but this isn't called the land of the blue sky for no reason.
Summer: June-August: Can get hot in the Gobi, but warm and moderate generally. Most chance of rain (*can snow/freeze in June at night), this is when the country is the most lush, green, and stereotypically beautiful. Peak tourist season.
Fall (Sept-Oct): September is a month of rapid transition from warm to quite cold. October is still "fall" but expect the first snow and freezing temps. The larch trees turn a bright gold and are very beautiful.
WInter: (Nov-March): COLD and dry. The air quality in UB is quite bad due to lots of coal burning, but out in the countryside this is the most beautiful time of year in my opintion. Just crystalline clear frigid weather. At the extremes it could get as low as -50 F or lower, and that isn't including wind chill. However, typically it hovers around 0 F with reasonably likely cold fronts dipping into the mid -20s F.
There are some mountains that have historic/religious significance and cannot be climbed, however barring those specific caveats you may want to consult a local guide service about, the majority of Mongolia is free-reign. As long as you aren't very obviously infriging on a designated location or clearly inside a fenced off area of property near existing structures, you are generally free to explore!
Lodging has a few options.
Hotels/Resorts: They exist and more and more are being developed, however in those remote places it is highly unlikely you will find one.
Ger Camps: These are more common, and can range from extremely rugged (no-frills ger, completely dark inside with dirt floors and an old rusty stove fueled with dried animal droppings) to quite luxurious (well-appointed oversize gers with windows, electricity, wi-fi, hand-carved furniture, high quality stoves with fully-attended stove service using actual wood or coal). These camps generally will have dining available too, included in the price.
Wild Camping: If it isn't fenced in or obviously part of an existing ger camp, pitch a tent! The options are nearly limitless.
Food/Drink: I will caveat that Mongolia BBQ is a fake western invention, and is absolutely not what you should expect to eat, ever. Monoglia is a very hard environment to live in, I would describe the cuisine as 'survival food'. The climate is not conducive to any fruits or vegtables whatsoever, not much of that. Some potatoes. Especially in the countryside, expect goat meat as the primary protien, as fatty as possible to maximize calorie intake, especially in cold months. Maybe chicken or beef as a rarity. The standard recipe for food is protien-grain-cabbage stew, for all meals. There are a few different grain+meat+spice combinations that usually are in some form of dumpling or pita-like wrap that come fried/roasted steamed or boiled. And that is about it for food, every meal, every day. Boiled-whole sheep head is special occasion meal. I will say that there is a traditional BBQ style (the real Mongolian BBQ) that involves burying lamb, sheep or goat in the ground with hot rocks, and it turns out pretty dang delicious. Smoky flavor from the smoldering organics in the dirt. Beverages; Vodka is the drink of choice, although there are absolutely some Mongolian beers. Due to the prevalence of free-range livestock, definitely purify all your naturally procured water. I used city and well water to do things like make coffee, boil pasta etc, but stuck to bottled or purified water for straight drinking. Of course, the fermented horse milk........This is mare's milk that is harvested in the winter and generally left to sit and ferment until the next spring. Personally......I did not like it. Alcoholic old milk just is not a typical western-palatte flavor we are used to, but hey, why not give it a try. Their normal non-alcoholic milk tea is quite delicious, I drank a ton of that stuff. In UB you will find a wide variety of foreign restaurants, some are pretty good. All together though, Mongolia is not a cuisine-destination.