Deerhunt: Welcome! You have come to a good place.
First off: You HAVE a 243. Its not costing you anything.Shooting any factory loaded ammo isn't cheap.
I should not make assumptions about your finances,but when I was 13 I could not afford a lot.
The Old Cowboy said "If you quit spending money on what you want,you can afford what you need"
But it sure is fun to think about these things,and we learn that way.
I know of a woman who gets her elk with a 243. I know of a guy who shoots really big mule deer with a .243. You would use a fairly heavy. and tough bullet. You don't just whang away at the animal. You be patient.The critter has to be still.You have to know where the bones and organs are inside the animal.
You have to see the heart or the lungs . And the ribs. You have to be steady and know for sure you are sending that bullet through those ribs into the lungs or heart.
If you don't have that shot,you wait or get closer.
You also have to know your cartridge,your rifle,and yourself. How well can you hit a grapefruit or a cantelope in a sitting position? At what range? Not much hunting is from a bench.
Before you buy a $400 rifle,I suggest shoot $400 worth of ammunition.
That's how you make a .243,or a 6mm,or a .257 work. In fact,thats how you make any rifle work.
You can make your own choice,its OK to prefer a pump. Just my opinion,if you key in on making the first shot count,the rapid fire thing is not important.
You can get a bolt gun in more cartridges,they are typically lighter and more accurate.
If you are going to have two rifles,a .243 and a .308 is a great team. If you are just going to have one, I'd consider a 7-08.(There are a lot of great cartridges,including the 270. )
I'm not trying to limit your capabilities,but I'd agree with waiting a few years to take up handloading.
I have a .257 Ackley Improved,soI know the .257 some.
Some outfits like Hornady might make some modern hotter loads with spitzer bullets,but the standard Win/Rem loads have always been loaded down for older guns that weren't as strong and might be home made.
And they put a round nose bullet in them that did not fly so well.Don't get me wrong,still a great deer cartridge!. But those 117 gr Round nose bullets from a .257 don't keep up with a .243. At 200 yds,It does not matter,the .257 makes venison.
As much as I like what I can do with a handloaded .257,I would not recommend one to anyone who did not handload.
Remington brought out the 6mm and it could have done better but Remington rifled it for varmint bullets. It didn't shoot deer bullets so well.The 6mm is another great cartridge for a handloader. The 243 is a little shorter,too. It works better in todays rifle.Ammo is easier. So the .243 is generally the better choice today. Its a lot harder to find 6mm. There isn't a nickle's worth of difference between them in the field.
Once again,shooting,practice,developing as a shooter is far more important than what the name of the cartridge is. You DO need to pick the right bullet.
There isn't much you can't do with your .243.
You spend all your money on guns and not enough on ammo and a 300 magnum might not work for you.
As far as shooting 600 yds??? Well,really,I suggest you just get real good at 200 or 300.
The longer the range,the more chance of wounding an animal. Then that animal is suffering,and you will feel awful.
Keep it a sure thing quick kill to the best of your ability.