If I were to Start Over...
If I were to start over as a hunter, and I only had a choice between a .243 and a 30-30, I'd be inclined to go with the .243 as a "first" gun. -But I'm not a beginner, and I started out hunting 'chucks with a Savage 340 in .22 Hornet. I got a Remington 760 in .270 as a high school graduation gift, and as soon as it was acceptable I got rid of it. The gun couldn't group tighter than 4-5" at 50 yards. Overpowered and underaccurate.
I have my Dad's remington Model 14 (.30 Remington) today; he's where he no longer needs it. The gun was bought in 1920 for $12 by my grandfather, who raised and fed 5 kids through the depression. For 2 generations, it was never fired without bringing home venison. Most of my Dad's hunting buddies, the men with whom I learned to hunt, carried 30-30's. They were all good hunters, and all had venison most years. So I will not disparage the 30-30 (or the .30 Rem - same ballistics).
My own 2 go-to guns are my .257 Roberts (Ruger 77) and my maternal granddad's 6.5x55 Swede sporter, now mine. My Roberts has done several whitetails, a muley, and an antelope. One bullet per animal, all DRT.
My young wife had never handled or fired a gun until a year ago. I gave her a Ruger M77 lightweight .243 as a first gun, and she carried it this past fall. She has "eyes" for my Ruger .270 Win. already, and who knows where she'll end up. But she started with the .243, and became very, very accurate with it. She can carry it all day, and she has confidence that she can put the bullet exactly where it has to be to do the job. That counts for a lot. Thus, my opening statement.