The best place to start would be the New Mexico Game and Fish web site

http://wildlife.state.nm.us/

As far as your other questions have you ever shot a rifle or have any experience around them?  Sometimes hunting can be just luck and other times you have to hunt your rear off.  The best thing to do is once you get moved out there is to see if anyone where you are going to work hunts and see if you can tag along with them if you draw out on a permit.  Then if you do get a permit and manage to find an animal to shoot and it goes down be prepaired for a lot of work.  An elk will be anywhere from 200lbs for a calf to over 1000lbs for a bull.  A rifle is a matter of choice, people use a lot of different calibers for hunting elk but the most popular ones are going to be from a 7mm Remington magnum on up to a 375H&H magnum with a whole lot inbetween.   

You can't elk hunt without a permit, and to get that you generally need to apply.  New Mexico typically has better deer leftovers than elk leftovers. 

When you begin hunting you will never look at spending time in the woods the same again.  You may feel like an experienced outdoorsman, but most hikers never leave the trail by a significant distance.  When you begin hunting you will frequently find yourself 3 or more miles from the nearest trail, which is rare for a typical hiker.

What I'm getting at is, try starting with a leftover deer tag in New Mexico just to get more experience in the outdoors if you can't find an elk tag in an area you're willing to hunt.

There is also a series of articles designed for the new person at Colorado's Elk Hunting University.  They kind of walk you through the whole process.

http://wildlife.state.co.us/Hunting/ElkHuntingUniversity

I also wrote an article outlining some of the differences between each state, including application deadlines and fees:

http://www.biggamehunt.net/articles/nonresidents-guide-western-hunting

Initially, it is very difficult to come up with some good advice in one paragraph.  I have been elk hunting for years and I learn new things every year, both on the hunt and in preparation.  I would recommend you go out and buy a book by Mike Eastman called, "Elk Hunting the West."  it is really good and it is accompanied by a DVD.  The book really explains a lot about elk hunting, from looking at maps and what to look at on the topo maps to narrow down areas to scout, what to look for when scouting, etc...  I got mine at Sportsman's Warehouse for around $20-$24 but I am sure you can get it cheaper on Amazon.com.  There is another book called Elk Hunting 101 that is also very good.  Other than that you'll have to get out in the field and give it a go.  Get some good binoculars, at least 10X42.  The more you spend the better the quality. You want an optic that lets in a good amount of light so that you can use them in the early morning and late evenings.  You will want to get some camo that is comfortable and quiet and preferably matches the terrain you are in.  I frankly think that King's Camo has some good camo patterns for the western states.  They also make some that are really comfortable and quiet.  I archery hunt and King's keeps me warm in the cool morning and cool in the warm afternoon.  A good boot is also a must.  I use Salomon boots personally and have loved them from day one.  They are a little on the pricey side but if you get them from the right place you will be able to get a good warranty.  They are light and made with Gortex so they keep your feet dry.  The weapon you choose should matter.  I used to rifle hunt and used a 30-06.  Then I graduated to a .300 Winchester Short Mag.  This carries a nice punch which provides for great accuracy at close range and long distance.  If I were you, I would not get a wimpy gun.  You will really limit yourself if you go for a gun of small caliber.  While the small gun can get the job done at a certain range, you will really make it more difficult on yourself.  When out in the field, the elk need both food and water and from my experience and from reading, they will not go more than a mile from either.  If you can find a good area with these then you'll have a good chance.  Be quiet, do as much as possible to reduce the amount of scent you leave, and keep a good lookout.

TMH