Fry/drive visitors can fly to Las Vegas, NV, St George, UT, or even Cedar City, UT before driving to Zion. Salt Lake City is still quite a ways from Zion, but it is an option since many carriers fly there directly. There are more options I'm sure, but the above are the major ways I can think of to lessen the pain of driving long distances.
Since the American Southwest encompasses a very large region (spanning multiple states), we'll try to answer the question of how to get here based on specific regions and based on a road trip leaving Los Angeles (as there are simply too many options to cover comprehensively in this website).
Zion National Park: This park sits at the far southwest corner of the state of Utah. Coming from Los Angeles, we'd drive the I-15 past Las Vegas, Nevada, and then for another 2 hours (along the I-15 then on UT 9) to Springdale, Utah, which sits right at the mouth of Zion National Park. Total driving time is usually between 6.5 hours to 8 hours, but it all depends on Vegas traffic as it's not unusual to see terrible traffic on the I-15.
Bryce Canyon National Park: This park is about 1.5-2 hours further from Zion National Park heading east via UT 9, then north on US 89 before heading east again on UT 12 (through Red Canyon State Park - an excellent intro to hoodoos you'll see in Bryce Canyon). From there, you can turn off south on UT 63 into the park.
Grand Staircase National Monument: This is a giant reserve area further east of Bryce Canyon along the UT 12 and southwest of Capitol Reef National Park (which sits further to the east near the junction of UT 12 and UT 24). The main town in this area is Escalante. There are regional flights that come here, but if driving from Bryce, look at around another hour east along UT 12.
Capitol Reef National Park: This very lightly visited park sits just east of the town of Torrey as well as the UT 12 and UT 24 junction. You're looking at another hours or so of driving from Escalante, UT to get to the western entrance. There's a lot of nothing to the east of the park (except for the odd town like Hanksville where there are accommodations as a pit stop).
Arches National Park and Canyonlands National Park (Island in the Sky District): Arches NP and the Island in the Sky District of Canyonlands NP are close to the town of Moab, UT. There are regional flights that serve the town, but if you're driving, it's over 4 hours of driving from Escalante (along UT 12, UT 24, I-70, US 191).
From Moab, it's about 15 minutes northeast to the Arches NP entrance. It's about 30 minutes or so northeast from Moab to UT 313, which eventually connects you to Canyonlands NP as well as Dead Horse Point State Park.
The Needles District of Canyonlands National Park is more of a backpackers mecca as most of the hikes are very long. Some limited (although hairy) four-wheeling is also allowed there. It's about 1.5-2 hours south of Moab, or a little shorter north from Monticello.
The Maze District of Canyonlands is the most remote district. You'll need to be self-sufficient and a hardly high-clearance vehicle to even make it to the park. It's as far away from civilization that you can get to in the lower 48 states and you'll have to come prepared as such if you're going to attempt it. Definitely allow at least a week for this. Entrances are from UT 24 (opposite Goblin Valley State Park turnoff) or from I-70 near Green River.
Natural Bridges National Monument: Right at the junction of UT 95 and UT 261. The small park sits south of Canyonlands NP and is pretty much in the middle of nowhere (2.5 hours south of Moab via US-191, UT-95, UT-275, or 4.5 hours from Escalante via UT-12, UT-24, UT-95, UT-275).
Monument Valley Tribal Park: on the Arizona/Utah border on US 163 between Kayenta, AZ and Mexican Hat, UT. Can pass by Utah State Parks of Goosenecks (turnoff US 313) and the Valley of the Gods along the way. Monument Valley is about 3 hours south of Moab or 6 hours east of Escalante.
Glen Canyon National Recreation Area: this area encompasses the manmade lake of Lake Powell. Page, AZ is the main developed area as that's where the dam responsible for the lake is located. Page, AZ is about 9 hours from Los Angeles via I-15, UT-9 (through Zion NP), US-89.
There's also a separate marina at Hite, UT, which is off the remote Bicentennial Hwy UT 95. Far less developed out there, it's also the less popular spot to be launching a boat into Lake Powell though much quieter.
Page is a good base for canyon adventures as well as aquatic endeavors. The Paria Canyon-Vermilion Cliffs National Monument is nearby as well as the Rainbow Bridge National Monument.
There's also the dramatic cathedral-like Antelope Canyon just southeast of town.
Grand Canyon National Park: there are two distinct regions of the park separated by the Colorado River.
The North Rim is accessible via AZ 67 whose turnoff is near Jacob Lake. The nearest major developments are Page, AZ (about 2 hours northeast via US89 and ALT-89 then south on AZ 67) and Hurricane, UT (over 2 hours southeast via UT59, AZ389, ALT-89 then AZ67). Hurricane, UT is between St George, UT and Springdale, UT on the UT9.
The South Rim from Los Angeles, is about 8 hours of driving. The highways involved are I-15, I-40 (turnoff near Barstow, CA), then AZ-64 into the Grand Canyon Village. There are long stretches without gas on the I-40 so make sure you don't take a chance at running out of fuel in the middle of nowhere are fill up when you can.
The Havasupai Indian Reservation is a side canyon of the Grand Canyon, but is watched over by the Havasupai Indian Tribe instead of the National Park Service. Getting here is quite out-of-the-way as you first have to drive to a trailhead, then hike or horseback or helicopter for 8-miles to the remote village of Supai. From Los Angeles to the Hualapai Hilltop (the car park and trailhead), it's about 7.5 hours nonstop along the I-15, I-40, AZ66 (the historical route 66), BIA18 (near Peach Springs). Given the remote nature of the drive and hike (tack on another 5-6 hours), some people actually sleep in their cars at the trailhead before hiking to Supai. In our case, we spent a night at Laughlin, NV then did a pre-dawn drive to the Hualapai Hilltop ultimately starting our hike at around 7:30am. For a place as remote as this, you'll be shocked to find that parking is not necessarily easy to find at the Hualapai Hilltop.
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