I'm as much (maybe more this time of year) into mountain biking as I am into skiing! (And that is saying A LOT). I race downhill mountain bikes, and Super D, and do a bunch of trail riding in Moab, and XC stuff on my new singlespeed around SLC.
But basically, you need to figure out what sort of riding you want to do. I would really recommend against a hybrid bike. That type of bike is basically not good at anything. If you want something that can handle both, I'd recommend a decent hardtail mountain bike. You can get a second set of tires (or a second wheelset if you're really into the road riding and want to make it even quicker), and then it will be decent on the road or goofing around town and good for riding trails. A hybrid bike would be horrible off-road unless you're just going to ride gravel roads, which - trust me - is no fun at all. If you want it to be fun, you need a bike that can handle some singletrack. That's where it's at in biking!
Just to give you an idea of the diversity in bikes - this is my bike "quiver":
The pink one is the singlespeed. It's fully rigid and has one gear. (What I'd recommend for you would have a front suspension fork and gears, but a singlespeed has it's advantages too - but it's generally not where you want to start.) It's very lightweight, great for climbing hills, weaving through twisty buffed out singletrack, but not so fun on rocky descents or more technical stuff.
The one in the middle is my trail bike. Most people go with about 5" of travel front and rear - mine has 7" in the rear and 6" in the front. Mine is heavier than average, but built to take some abuse. We like bombing down rockly ledges and stuff in Moab and this can take just about anything, but is geared to climb up to get there on your own.
The one of the left is my downhill bike. This has 8" of travel front and rear and no front derailleur (it has gears, but not as many as a trail bike and it's geared to just pedal fast on flats or downhill). It's nearly impossible to pedal uphill on even a small incline. It weighs about 42 lbs, and is made just for racing DH when you've got lifts to ride, can shuttle, or just want to push the dang thing up to ride back down. It's also made to handle big drops, rocks, etc.
As a skiing comparison it's like a superlight AT setup (backcountry XC skis), a burly AT setup, and big fat skis with alpine bindings for riding the lifts. A road bike would be more like an XC skate/race setup.
What you should get really depends on how much you have budgeted to spend. I can think of some good starter bikes to try, but maybe you'd be better looking for something used.
Which bike shop I'd recommend also depends on where you're moving to exactly within SLC and what type of riding you want to do. Go-Ride.com is my main shop, but they focus on DH/Freeride type stuff. Canyon Bicycles in Draper is excellent for a wider variety of bicycles (from road to freeriding).
As far as road riding goes - I personally think it's fun but won't do it. Traffic scares me. My husband was nearly killed by a driver who fell asleep and flipped his truck in our residential neighborhood last year. I'll jump off 8-10' cliffs on my bike but won't ride the road much at all. On any off-trail ride - if I wreck - it was my fault. It's my choice to try something difficult or walk it. On the road, I'm at the mercy of bad drivers who I can't outrun. That's just not for me.
Hope that helps and you get into it - it's a blast!! :D