What follows is meant to supplement some of what has already been written, and endorses an emphasis upon technique as primary, acknowledging significant gains can be made with appropriate high-specificity exercises. This answer relates to punches only, and describes transformative techniques I was taught by a student of biomechanics.
Jabs
Instead of trying to push your fist forward, instead imagine there's a rope tied around your wrist and that someone is pulling your fist away from you, towards the target.
Why it works (for me).
The jab is typically utilised as a fast, rangy punch; as a range-finder, a deterrent counter and as an outright offensive weapon. When beginners try to infuse this punch with power and speed, they tend to do at least four things wrong which hamper its effectiveness. A) They tense up, which slows the motion. B) They retract the punch before throwing it, which telegraphs the punch and almost doubles its execution time. C) They don't rotate the shoulders, losing invaluable range and considerably restricting power. D) They don't punch through the target, greatly reducing the impact of the strike.
When I began to imagine my fist being pulled forward, it became easier to keep my body relaxed until the moment of impact (increasing speed dramatically), the temptation to 'chamber' the jab disappeared (so my punches became far more difficult to read) and my punches grew longer (maximising my effective range and making it easier to avoid my opponent's attacks). At first, I neglected to punch through the target, but by learning to focus about a fist's length beyond the bag, my jabs soon grew much, much faster and heavier. They key is to tense at the moment of impact so that you maintain acceleration for as long as possible prior to impact, but simultaneously ensure your wrist is rigid enough upon hitting the bag to transfer your body weight properly 'through' it.
Another tip: Work on doubling and tripling your jab, being sure to retract it properly to your jaw every time, as quickly as possible. Studies have shown that training the negative motion of an explosive movement is highly beneficial to the positive motion. When you repeatedly throw the jab, the retraction/forward bounce effectively becomes a form of plyometric training which has long been recognised as one of the primary methods by which to achieve athletic explosiveness. Retraction speed of course also allows you to regain a defensive posture more quickly and increases the potential speed of your combinations.
You can add very light (1/2 - 1 kilo to start) dumbbells when shadow boxing if you wish, but exercise great caution with higher-resistance plyometrics. It is very easy to injure yourself and elbow hyperextension is a real danger. A safer alternative can be to use a loop of lightweight resistance band. Hold one end of the loop in each hand and wrap it around your back so that when you punch, the band adds resistance. Key to this form of training (or any highly motion-specific exercise when added resistance is used) is the maintaining of proper form. It is easy to let tired muscles fall into bad habits, and for your hard work to translate into the reinforcement of these bad habits, and into injury from poor form.
Crosses
The same principles and drills/exercises apply here, but there's one very useful and under-utilised aspect of technique which comes into play when transitioning between hands during combinations.
Imagine you are attached to the ground via a pole which extends through the top of your head, down through your neck and torso, to between your feet at the ground. In ideal circumstances, that pole represents the axis point around which your shoulders rotate when punching.
Many beginners throw one punch and retract it before launching with their opposite hand. Whilst this can be appropriate if during a fight you are throwing sporadic punches, if maximisation of power during combinations is the goal, the retraction of one hand should be accompanied by the forward thrust of the opposite hand. This is where the pole comes into play. Imagine throwing a jab with full extension, so that your shoulders rotate around the pole to be parallel with the line of the punch. Now, as you retract that punch, you will notice that you can allow your shoulders to operate relatively passively, rotating around the axis to throw a long cross, which is what many people do, emphasising the arms instead of the torso, losing a lot of power in the process. Instead, imagine pulling back firmly with your jab-side scapula and shoulder as you punch forward with the cross. You are (almost) doubling the force applied via the rotation of the shoulders when you do this, resulting in a quick performance gain. Note the aforementioned exercises with light dumbbells and resistance bands apply to crosses as well as jabs.
