Do NOT go to Brian Head. If you have an opportunity to go to UT, you def want to be in the PC/SLC area. I'll try to give a condensed overview here. Also, the search bar is your friend. There a ton of threads similar to this that have more info.
If your main focus is shredding pow, the answer is definitely Little Cottonwood Canyon. Snowbird and Alta have the best combination of sick terrain and tons of snow. It's $69 for an adult day pass at Alta and $74 for an adult Tram/Chairs pass at the Bird. You can also get lower pass prices if you look into multi day tickets and AltaBird passes are also available (you can find more info on their websites). Neither mountain has a park, but it really doesn't matter considering how much varying terrain and natural features are available and Wildcat lift at Alta (along with a number of other areas at both resorts) is an all natural terrain park that on a pow day is more fun than just about any man made park. Snowbird tends to be faster paced and caters more to tourists whereas Alta has a more local, core feel. There are certainly nice accommodations and good food if you stay in LCC, though there won't be a whole lot in the way of nightlife. Just do some research on the hotels/lodges and figure out what fits your style/budget. I think there may be some XC up there, but don't quote me on that and if there is it is certainly limited.
Big Cottonwood is sort of a little brother to LCC. It contains Brighton and Solitude which can both be very fun mountains, though they may fit your criteria the least. Passes at these mountains are the most affordable in the area with Brighton being $62 per day and Solitude $68 (again check their websites for multi day deals and info on SolBright passes). Brighton is more of a locals hill than Solitude and caters more to the younger crowd. There is a park there which at times can be fun, however it is nothing special. The jumps are pretty much never good, however there are some fun jibs, especially the lift tower rail line. Brighton can be really fun on a pow day, just don't venture anywhere without guidance from a local, it's easy to get cliffed out and into real trouble if you head into Rock Gardens or anywhere outside the resort boundary (this obviously applies to all the mountains, ropes and signs are serious business in UT and anywhere out west). Solitude is more family friendly and offers more lodging options along with a pseudo "resort village". Both mountains are smaller than their counterparts in LCC, however they can still be very fun on a pow days especially if you're a relative newbie to REAL mountains with REAL skiing in REAL powder. They don't receive quite as much snow as Alta or the Bird, however they still routinely have 500+ in. seasons and the slight lack of snow (which you really won't notice in the course of a one week trip) is made up for by smaller crowds and more fresh tracks on a pow day. Don't expect anything in the way of nightlife at either resort and there is only minimal XC access at Solitude.
Then there is Park City. PC blows BCC and LCC out of the water when it comes to food, accommodations, park skiing, and XC. Where it Park City lacks is cost and pow shredding. Everything costs more in Park City. From food, to lodging, to lift tickets, it is all going to cost more.
There are a TON of restaurants in downtown Park City as well as the surrounding areas and at the resorts base areas. There is something for everyone from affordable (though still tasty) pizza joints that cater to locals, to high class, over priced restaurants that are good enough for the Hollywood elite when they come to town for the Sundance Film Festival. Accommodations are the same way. Again, there are a ton of options from affordable (this term is all relative) hotels in nearby Kimball Junction and the outer parts of Park City, to places in the resort base area, to ski-in ski-out condos and houses for nightly rental that most of us could only dream of being able to afford. The nightlife also has variety featuring sports bar/grills to local pubs to some of the most well known bars and clubs that ski towns in the U.S. have to offer.
As far as the mountains, Deer Valley really doesn't have anything in the way of park. It tends to cater to the wealthy recreational skier, but the average skier can still have fun. Because many of the skiers there are recreational, there is plenty of fresh to be had on a pow day and a surprisingly large amount of terrain to explore and get fresh tracks on. DV is the most expensive of the 3 PC resorts with an adult day pass going for $90. Park City Mountain Resort (PCMR) probably caters to families the most out of the three Park City mountains and consistently has one of the best parks in the nation along with probably the best pipe. People often hate on the pow/all mountain skiing at PCMR, however it is better than it is given credit for. Sure, it can't come close to what BCC or LCC has to offer in terms of terrain (both quality and quantity) or snowfall, but if this if your first time out west and you have skied much in the way of steeps/drops before, you will still have a damn good time. Check out McConkey's, Jupiter, Thaynes, and Pioneer lifts on a pow day. PCMR truly has something for everyone and is located closest to the heart of the town of Park City along with a lift that goes right into the town. This years ticket prices haven't been listed yet, but last year I believe they were $81 or $82 per day and will likely be a few buck more this year. The Canyons has the most acreage of the PC resorts along with the most steep/challenging terrain along with a well rounded, fun park with good flow. The park is also usually less crowded and more laid back than PCMR. The Canyons has an open gate policy meaning you can venture out of bounds whenever you like at your own risk. Unless you have a beacon, shovel, probe, sufficient knowledge of the backcountry and avalanche awareness, and a partner with the same, do NOT go out of bounds. Despite being on the frontside of the mountain, avy control is not performed on these areas nor are they patrolled. Tons of idiots with no knowledge or gear go O.B. in these areas at The Canyons, that does not mean that you should. There is a ton of fun terrain in bounds at The Can and you will not be able to ski it all in a week. Pass prices have also not been listed here although they are comparable to PCMR.
This is honestly a pretty abbreviated overview of the PC/SLC ski scene, again do some more research of your own on the resorts websites and by searching for similar threads. There is a lot more that could be said about all of these areas, and despite living there the past 3 winters, I know I am not the most knowledgeable person on this subject and there are a lot of other NS'ers that can provide more specific knowledge/insight. Bottomline, do your research and find out what area fits your needs the best. If you are able, the best option is probably to spend a few days in each area (PC, BCC, LCC).
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