You've received a lot of good advice already. If you're looking at Idaho - this web site will be of considerable help: Idaho Fish and Game Home
I believe all the western states have similar web sites with license fees, season dates, GMU boundaries etc. If you dig a little deeper, you can find important stuff like success rates...
An outfitter is a great way to go - IF you find a good one and IF you have the money. Otherwise, public land, do-it-yourself elk hunting can be a very "iffy" thing. Maybe you'll get on them, maybe you won't.
Your .338 rifle has more than enough punch. Looks like the rifle is a done deal, you're good to go. My only caution would be to be prepared for a more traditional shorter-range shot too. Elk can be goofy. Maybe you'll be all set, watching a meadow a half mile away, and then a bull stomps out of the tree line 50 yards away... Yikes! Or one goes into thick "black timber" and the only way you're getting him is to go in after him, hunting slowly... You may want that long range shot, but I'll bet you'd rather go home with a big ol' 6x6 filling that tag...
BTW - most elk taken don't look anything like the big trophy class bulls shown in all the hunting magazines... Many of us just shoot spikes, cows, raghorns or youngish bulls... But an outfitter will understand that you're looking to go home with a trophy and will work to make that happen.
Be prepared for harsh weather, even in the early fall. Doggone elk often live and are hunted up at 9,000, 10,000 or even 11,000' in the western states. In Wyoming we found ourselves at nearly 11,000' once, and I tagged my bull at about 9,500 - 10,000' as I recall. Being in shape helps, as does getting up to high altitude a few days before the season, to acclimate. Good clothes, and a good warm sleeping bag are very important.
I hunt elk here in Washington, on my own or with friends/family. Took a real nice, wide 6x6 some years ago in Wyoming. The shooting wasn't a problem, but it was only about a 180 yard shot. All your whitetail hunting experience is going to pay off in a big way. Patience. Marksmanship. Understanding that game animals often don't stick around and offer you a perfectly still, broadside shot. All of that will help.
Hunting for elk in Washington last fall. Didn't fill my tag, saw a huge 6x6 bull... During deer season!
My Wyoming bull, about 10 years ago:
I like that you're preparing so far out. Good call. Maybe see about taking a trip "out west" on vacation during the summer to see some elk & elk country.
Best of luck! Guy