the Hushed Casket

I apologize if this is an insulting question, but did you zero your scope? You can't expect it to hit the target if you don't adjust the crosshairs onto the target.

It also sounds like you're having a problem with mounting your scope. The scope base (if one is used) and rings are extremely important in holding a zero. Take extreme care in making sure the metal surfaces mate correctly. Use blue locktite on all the screws. Cinch them down tight, but don't strip the threads. Then go zero your scope. Start at 25 feet or so and get onto the target. Use the click adjustments for elevation and windage until you get a 5-shot group that is centered on the bull. You can calc how many clicks per inch you need at a certain range, but with inexpensive scopes the stated figures never hold true, so just go with trial and error. One piece of advice on zeroing: be bold in your scope adjustments. If you try to "sneak up" on the bull you'll be zeroing all day. Move the scope a bunch of clicks when you first start out (be bold with adjustments). If your group moves from 3 inches right of the bull to 2 inches left, then you know about how many clicks to go back right. This is called bracketing. It's a better way to zero than "sneaking up on it."

Go with whatever caliber rifle suits you. .308 is one of the most popular calibers in the world, not just among Marines, so if it's not prevalent in your area then it's just your local area. .308 Winchester is arguably the most documented and widely loaded precision rifle caliber in the world. It is essentially a shortened 30-06 round. It was standardized as 7.62 Nato and used in military weapons by over 100 countries from the 70s until present. Surplus ammo is available for $.12 per round. However, if 30-06 suits your price and availability, then go for it. It's also a fine cartridge. It's the cartridge .308 was derived from, IIRC. You'll just pay more to shoot it, get more recoil from it, and unless you hunt Elk or larger you won't notice a difference in its ability to take game.

The biggest name in shotguns is the Remington 870. Its main competitor is the Mossberg 500. I've owned both and I like them both. I like the parts availability and customization opportunities of the 870. I like the safety location on the M500 better. The M500 is about 10% cheaper. It is also more difficult to thoroughly clean. Handle both and see which one suits you. They are both excellent scatterguns. I would suggest a retired police shotgun with a beat-up wooden stock. I bought one for $130 several years ago and turned it into a great bird gun (swapped the 21" barrel with cylinder choke for a 27" barrel with modified choke), and gave it a homemade Krylon camo-paint job.

I would suggest starting with a pump because they are so much cheaper than autos. After that you can explore autos if you like shotgunning. For autos I like the Browning Gold series, but they are pricey. You won't find a softer recoiling 12 GA. Remington 1100 is the most inexpensive way to get into a high quality auto, IMO.