That's really going to depend on alot of factors. Do you want a trophy, or any elk? Do you want to spend alot of money, or limit it? Do you want to hire a guide, or do you want to do it yourself? Also, other factors will play into your decision. Certain states/areas have strictly lottery for elk, and are real long shots at getting a tag. Others have plenty of OTC tags available.
I would say, start with Colorado if you can do it. There are multiple seasons, liberal tag quotas that can even be bought over the counter, depending on season. You'll have the opportuniy to hunt cows, spikes, or big bulls if you pick and choose correctly. Finally, CO has alot of public land available to hunt. Alot of hunters on it, but if you are committed, you can make it work.
There are plenty of Colorado elk hunters on here who can get even more detailed than I can. I would talk with one of them.
Since you have never hunted elk before that means you have no prefernce points accumlated. In my opinion and not just I cause I leave here, CO would be my top pick of where to go. We have alot of public land and we give away alot of tags each each. There are alot of unit that hold good elk numbers and do not require and PP or you can get a tag over the counter. CDOW webiste has alopt of good information on where to hunt and what it takes to get a tag. I can try to answer any question you may have just feel free to ask. 1st thing you have to do is try and decide what method you want to hunt with bow, rifle or muzzleloader, now you know what you will be hunitng with. Next comes what area you want to hunt in, that can make or break the hunt. Pick an area and then get as much info on that area as you can via, internet and DOW's and if at all possible go out to that area in the summer to scope it out as far as camping and acess. Better to realize it was not how it looked on a map months before the seasoln compared to opening morining. Last but not get be in somewhat shape...elk hunting is tough on the body.
Like others have said, we really need more information from you to give you a good answer. With out knowing it all the others have given you solid advice. Colorado has an over the counter bull tag for both the 2nd and 3rd combined rifle seasons. I would highly suggest getting onto the Colorado Division of Wildlife's website and scouring it. They provide you with absolutely everything that you would need to plan a hunt to this state. There is a new section called Elk Hunting University that walks you through the process of getting an elk... from the point that you are in now to the shot. It is a very useful tool for hunters that are new to the western hunting game. Also check out the statistics section of the site. It allows you to look up harvest statistics for years past. It is very detailed and a lot of good information is stored there. You can also look up drawing statistics to see what your chances are of drawing specific tags. If you give the division of wildlife a call they can also help you in your search. Get started now if you are serious about getting out here. It is not going to be cheap but it will be worth it.
Kentucky is a place that one doesn't think about elk normally, but they have an up and coming program and I believe it is only five dollars to apply, non-refundable. Still you don't have to lay out the big bucks hoping to draw.
I plan to put in for a Kansas elk tag next year. Low odds of drawing though.
I learned this the hard way last year. If you're hunting in an area with lots of bear activity, it's a great idea to stash your food and water up in a tree. I like to wrap everything in a tarp and tie it between two trees. Don't just tie it up in one tree but tie it in between two trees at least 15 feet in the air. I thought I had my backpacking camp's food supply stashed pretty well last year. I returned to camp during the first weekend of archery to find all my food destroyed. Make sure...