While you said you've been shooting Dad's guns, we don't really know how much shooting you've done or to which cartridges you've been exposed to. Never the less, you can't go wrong with a cartridge having a bullet diameter between 6.5 mm and .30 caliber when combined with a case of moderate powder capacity.
The .270, .280, and the .30/06 followed by their ugly step children the .308, 7-08, and the .260 are all effective on medium to large big game over normal hunting ranges. Shooting out to 500 yards in calm conditions is so easy that you shouldn't be surprised to find yourself wanting more challenge from them. Of the 3 the .30/06 is my all time favorite, but according to Townsend Whelen, for whom the .35 Whelen was named, "the .280 is just a little better than the other two, but not so much better as to warrant trading one of the others to get one." The selection of 7mm bullets is nearly as good as the .30 caliber selection, the .270's selection is not quite as good, but still covers all the bases.
If you intend to handload, it doesn't really matter which cartridge you choose; you can make it dance to the tune of your choice, but if you restrict yourself to factory ammo, it will cost much more to shoot enough to become competent, and the full versatility of the cartridge cannot be realized. With handloads you can load for small game and birds ammo that is little more destructive than a hi-speed .22 for camp food or inexpensive plinking, match ammo for target shooting out to a half mile or beyond, varmint loads to trim down the coyotes, not to mention ammo that is suitable for all North American big game, and for 90% of the game elsewhere.
Choose the rifle and cartridge that grabs your imagination. Gun choice is a reflection of our personalities, but in your position I would get a Winchester M-70 in .30/6 and top it off with a low power variable scope, say a 2.5-8X Leupold. To me that typifies the big game rifle. It may exceed your budget slightly, but you'd have a top notch outfit to start you out.