I concur with the lower power scope with a modest objective diameter so that it can be mounted as low as possible.
Being able to get right on target with the scope starts even before you spot the target. The mantra you should follow is "eyes, muzzle, target". When you're scanning for game, your muzzle should track right along with your eyes; don't have your eyes looking one way and your muzzle pointing another way. That way, when you do spot something, your eyes and muzzle will already be lined up with the target, minimizing the amount of movement needed to bring the sights into alignment.
Keeping both of your eyes open and on the muzzle and target, the next trick is to simultaneously push the muzzle straight towards the target while pivoting the rifle up to your shoulder with the muzzle as the center of rotation. In other words, the muzzle doesn't go up or down, just straight out, as the buttstock swings through an arc up to shoulder level.
Then you pull the rifle straight back into your shoulder and achieve your cheek weld with the comb at the same time as the recoil pad hits your shoulder. If you do everything right, the scope will "blink" on and the crosshairs will be on or nearly on the target.
By the way, you should also take off your safety as you pivot the rifle up, and touch the trigger at the same time the recoil pad hits your shoulder, your cheek welds to the comb, and the scope blinks on. That way you're instantly ready to make any fine corrections in alignment and take the shot.
If your shooting eye isn't strongly dominant, you may have to "dim" or partially close your non-shooting eye as the scope blinks on, but not before. Then as soon as you take the shot open both eyes again and use your peripheral vision to follow the game as you reload with the rifle still in shooting position in your shoulder, again keeping eyes and muzzle tracking together with the target.