I shot my first elk almost 60 years ago when I was in college, and I used a borrowed .30-40 Krag with iron sights. As that was only my second hunting trip, I didn't know much about hunting, and I started shooting at him when he was so far away that the front sight bead completely covered him up. Luckily there was snow on the ground, very few other hunters around, and after a very long chase I finally got him.
The next year I decided that I needed my own rifle so I asked a couple of the locals that I worked with and had grown up elk hunting, what caliber rifle I should get. They both said to get a .30-06 or .270 Win, so I got a .30-06 with a Tasco then later a Weaver 3-9x40 scope. Except for a 3 year stint in the Army, my .30-06 kept my freezer full of elk meat for the next 10 years.
Then, just because I liked the looks of the cartridge case, I had my .30-06 rechambered to .30 Gibbs. That rifle shooting 180 gr Nosler Partitions, put another 20 some elk into my freezers, most with one shot kills, and none shot much over 250 yards.
Over the years I've also shot elk with my 7mm Rem mag, a 10 yd spot and stalk shot with an arrow, and one of my best 6x6 bulls with a 117 gr Sierra GameKing bullet from my .257 Ackley.
My favorite elk rifle now is my .300 Weatherby Vanguard shooting 180 gr Barnes TTSX bullets. I now have a Leupold VX 3i 4.5-14x40 CDS scope on it. I've used that rifle on multiple African hunts, and on hunts in New Zealand, Azerbaijan, and Canada. I see no reason for me to ever change to different elk rifle than my 7mm RM or my .300 Wby.
If I was just starting elk hunting, I see nothing wrong with a .30-06, 7mm RM, .300 Win, or the new 7 PRC or 30 PRC cartridges. I currently have 3 Weatherby Vanguard rifles that will consistantly hit the 9" steel gong we have at 430 yards at our range. I also have Leupold scopes on most of my rifles, and I sepecially like their VX 3i 4.5-14x40 CDS model, and maybe the one with the lighted center dot.
My best "trophy" bull elk in my younger days with my .30 Gibbs and an 180 gr Nosler Partition bulet.