Atroxus,
From a guy who taught himself to hunt, recommend you listen to what these guys are telling you.
If you bag a mule deer, you will soon learn that just moving the animal's carcass around by yourself is challenging, rolling it over, etc. - and a mule deer weighs perhaps half what an elk might weigh (and is much smaller). Gutting it, physically manipulating the carcass if you're going to hang it to skin it, hoisting it up, dragging it, all these things are interesting by yourself, because its between 300 and 400 lbs deadweight. By yourself, (especially when each step is something you've never done before,) it will wear you out.
Aside from the fact that in WA an antlered elk tag can be tough to come by, (and spike bulls are only slightly more plentiful than unicorns,) the weather is often a bit less snowy during deer season. Consider starting with a deer first, then move to elk.
About the mountain bike. Have you done much hiking or backpacking where you intend to hunt? (Will you be in the Blues, or over by the White Mountains?) Recommend you spend quite a few weekends this spring and summer backpacking in the area you intend to hunt, to find the water, locate the food sources, and get a feel for the lay of the land. This will also give you some idea of how far in you want to go, if you'll be pulling 800 lbs of meat back up the way you came. By all means, check out the area where you intend to hunt, but where I hunt the slopes are far too steep to think about trying to roll a bike either up them or down them. (I hunt in the Blues, in the extreme southeastern corner of WA.)
Like they said, the process is pretty simple - but the devil is in the details. You bone out the meat, some guys put it in muslin pillowcases, some wrap it in cheesecloth, pack it in a canvas bag, and then lash the bag to a packboard of some type. Then you spend the next 2 or 3 days getting the elk out (depending on how deep it is in one of the ravines...somehow they never fall "up" to the top of the ridgeline!) You generally hike down in the AM, load up your packboard, hike up, have lunch, hike back down, load up, hike back up, have dinner...x 3 days. Take it from me...this process goes faster with 2 or even 3 guys humping out elk meat. (If you've gone on numerous backpacking trips to get ready for hunting season, carrying something heavier than a 45 lb pack, then you may have some familiarity with the routine!
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There may be guys who drop an elk where they can drive up to it on 4-wheelers..., but I've not seen them. There ARE outfits that use horses, and these work well - but horses are expensive, one way or another. If it's just you, and you're lucky enough to bag an elk, you'll be humping meat for a few days.
(And I also recommend you look around for someone else to go out with - even if your buddy does not actually hunt. Some guys just like to get out for a few days, maybe cook some, sit by the fire, take photos, etc. Worth considering. If you break your ankle a few miles back in there, getting out could take you awhile. Be sure you file a flight plan if you go solo.)
If you practice your processing on a deer first, you'll learn a lot of valuable lessons when it comes time to pull an elk out.
Best to you, whichever way you choose to go.
Doc