Downhill is a looong way away, don’t even think of trying it until you’ve spent a few years either riding tamer stuff or in motocross. I’ve been riding for 15 years and I still get scared by DH runs. I like my personal safety a little too much.
I’d recommend looking into a 27.5” or 29” hardtail bike as your first mountain bike, which will probably fall under the Cross-Country (XC) or trail genres. Get a bike with a front fork, to get used to how suspension helps you, but don’t get a bike with a rear shock yet. A hardtail is a better teacher, as you learn how to move your weight and use your legs to absorb bumps and trail chatter. Once you’ve mastered that, a full-suspension (FS) bike can take you to the next level, but starting out on a FS is a good way to get in over your head before you’re ready.
I’d recommend the slightly larger wheels of a 27.5er or a 29er, as they roll over obstacles easier. Larger wheels means a lower angle of attack, which means you’ll also have more momentum to help you get over rocks, roots, and other things you’ll find on the trails.
If you can afford them, I highly recommend hydraulic disc brakes, as they provide improved stopping power and better braking performance, and it’s always a good idea to have a great set of brakes to back you up when learning.
As for budget, I’d probably be looking to spend between $1000-2000 on a bike. I know that may be a lot to swallow, but you want to buy as much bike as you can afford. Bike companies can buy components much more cheaply than you can, so it’s better to buy a better packaged bike than to buy a cheaper bike and upgrade components.
Keep in mind that weight plays a role as well. My first bike was a 31-lb hardtail Kona Fire Mountain cross-country bike with 26” wheels and 3” of fork travel. My current bike is a Giant TranceX 29” FS bike with 5” travel front and rear, and it weighs 29 lbs. I’ve got a lot more bike now, and it even weighs less! The trade-off was the cost: $500 for the Kona and $2800 for the Giant. However, it’s worth it; a lighter bike is easier to get up the climbs, flick around on the trail to pick the best lines, loft the front wheel over obstacles, and it all-around makes riding much more fun. Now that doesn’t mean to count grams like some roadies do, and carbon for a mountain bike is awesome but at minimum is $4000, so stick with aluminum. Keep in mind that this is a bike for the woods, and speed is all relative. I did a ride in the snow today where my average speed was 5.5mph, essentially a jogging pace. I still had a blast!
Another good thing to do is go to a few local bike shops and ask them what they would recommend for a beginner mountain bike. Keep in mind everything I just described, and ask a few different places to ensure that you get a variety of responses so that one shop doesn’t try to fleece you. Good bikes in the range I have described include the Giant Talon, Specialized Crave, Rockhopper and Hardrock, Cannondale F29 and Trail, and the Trek Superfly and X-Caliber. Your local shops should carry at least one of these brands, as these are the Big Four in the industry, but lots of smaller bikemakers produce quality bikes as well.
Hope this helps, and feel free to ask any other questions!