This turned out to be a bit longer than I intended, but I hope some out there are able to benefit from it. I'm taken by one thing in Noel's pics, and that is the number of peep sight equipped rifles. Those boys knew a thing or two about shooting. In those days folks didn't often spend money on stuff that didn't matter.
The .30/30 and the rifles that are most commonly chambered for that cartridge have much to offer. Lots of old timers cast 180 gr bullets and loaded them to what we know today would of been about 1800 fps and killed truck loads of stuff, large and small. The rifles are light and handy, but they are also surprisingly accurate. I have seen more than one 336 or M-94 shoot MOA at 100 yards. That's pretty good considering these rifles don't do much that the accuracy gurus say you have to do to achieve those results. Consider how much time effort and money has been invested over the years to bring about a bolt gun that is light and handy, yet all along the Winchester and Marlin carbines already had those attributes. Those rifles today continue to have the same advantages that they did at the beginning of the 20th Century; they are light to carry, they hold lots of ammo, they are accurate, and they are powerful enough.
Let's consider the powerful rounds that are commonly used for back-up against African dangerous game. Almost universally these rounds fire bullets at about 2100 fps. Regardless of caliber which begins at .45 or bullet weight that begins at 500 grs, the velocity sought was 2100. The magic of 2100 fps is that the bullets penetrate deeply in a straight line and are not likely to fail. The .30/30 shares that magic velocity. It was not designed to kill animals whose weight was measured in tons, but that was never it's job. It will handle game up to 1000 pounds quite well though. That 2100 fps velocity ensures that the bullet will penetrate deeply without failure. What happened when the .270 and later the belted magnums appeared? Bullets began to fail due to the high velocity. This was never an issue with the .30/30. It just continued to kill game with boring regularity, and without blowing 20 pounds of meat to mush. When bullet manufacturers began to produce tougher bullets to withstand the higher velocities of modern rounds, failures with the .30/30 began to come to light. It seems that when tougher bullets were loaded to only 2000 fps that they would sometimes not expand, and what was advertised as a better bullet, proved to be a poor choice for the .30/30. For a time, say from the '50's through the '70's the .30/30 caught a lot of bad press, and many of the "experts" disassociated themselves from it. There is a lesson to be learned here. If it ain't broke, don't fix it. The .30/30 loaded with traditional jacketed bullets with lots of exposed lead killed well, as did the cast bullets. They will today.
If you want to hunt with the .30/30, it is no different than any other big game rifle. It has a reasonable expectation of killing a big game animal with a single shot, from any angle, within it's appropriate range. The size of the game determines the range any given shot should be taken from, but in the case of deer and small bears, your ability determines what that range is. There are guys that should never shoot beyond 25 yards, there are other guys who can extend that 10 times.
You have to make the investment in time and ammunition in order to determine what you can do with your rifle on demand. What you did from a bench rest once doesn't matter. How far can you hit an 8.5"X11" piece of paper off hand, from kneeling, sitting , and prone? When holding the rifle in both hands, from 25 yards, how close can you snap shoot a pair of shots at a target with no distinct aiming point? Don't worry about shooting groups, worry about how far your bullet holes appears from the intended point of impact. Once it has exceeded 4", (meaning that it could be 8" between bullet holes) you are shooting at a range that is beyond your ability given the conditions present on that day.
If you are unimpressed with your performance, how can you improve it? If you are relying on the factory sights, the first step might be to exchange them for a sight that covers up less of the target. A ghost ring and post, a low powered scope, or even an electronic sight are better choices than the semi-buckhorn and bead factory sight.
The second step is more difficult. You must master the basics of field marksmanship before you will shoot well, regardless of the rifle you choose. This can be accomplished with weekly visits to the range supported with an hour of dry firing each day for a month prior to the season. Dry firing must be done with a goal in mind and in a way that will benefit you. When the hammer falls, how far have the sights moved off target? Learn to call your shot. Can you balance a quarter on the barrel near the muzzle and dry fire without dropping the coin? Keep records of your training and of your strengths and weaknesses. Don't ignore working the areas in which you are strong, but spend time on the areas that need work. If you can, work with a coach to prevent reinforcing bad habits.
Learn how to attain and check your natural point of aim, and get used to what that should feel like. If you close your eyes, relax and upon opening your eyes find that the sights have swung of target, you need to adjust your position relative to the target. Ensure you have a good cheek weld. If shooting with irons, focus on the front sight to the exclusion of anything else. Press the trigger, don't jerk, slap, or snap it regardless of slow fire or rapid. Breath between shots. Concentrate on the shot you are making, not the previous or the subsequent shot. Follow through, which means to remain focused on the front sight after the hammer has fallen. After the shot, work the action without getting out of position or removing the butt from your shoulder, and quickly reacquire your sight picture. In the field, conditions are in a constant state of change, so you are always shooting against the clock, train accordingly.