Rifts, D20, D10 and How to Run A Good Rifts Game.
1. The infamous -10 to strike, parry and dodge rule. Palladium is right in that it's damn hard to dodge a laser or a gun, or even a rubber band. However, Rifts is cinematic and epic enough that one could easily imagine Juicers, Cyber-Knights and Crazies dodging bullets in Matrix style (read some of the stories throughout the books, say, the Sir Taloquin story in Cyber-Knights or the Juicer stories in Juicer Uprising). Further, even a rookie could dodge a bullet not by actually moving away from the projectile when it was fired but by anticipating when the other person would shoot and leaping out of the way. For this reason, I instead make the penalty to dodge high-speed projectiles dependent on a few things. If a character is caught off guard or cannot see the weapon (say, if someone has a Derringer hidden in their hand while they reach out to offer a handshake), then the -10 without bonuses rule does apply. If the character can see the opponent and accurately predict when they fire, then I simply make it a -2 or perhaps -3. If the opponent is a bit more experienced and can avoid telegraphing their motion, it may become a -5. And so on. Simply think, "Would it be logical for the player to react somehow to the gunfire?" After all, if a character who can leap thirty feet high and twenty feet across standing still can predict when her opponent's going to fire, she can easily jump onto a nearby building, over her opponent, through a window, etc.
2. The melee attack system. The logic behind individuals getting a different number of attacks per turn is fairly realistic based on martial arts experience: Someone who has more experience in launching more attacks (i.e. has more melee actions) will get a few extra blows while her opponent is still figuring out what to do next if only for a few seconds. If you watch boxing matches, you'll see that oftentimes the more experienced fighter will get a few extra blows in before the bell or while the other fighter can only hide behind her gloves. However, this doesn't mean that someone with fifteen melee attacks is going to trade punches with someone with three melee attacks and then get twelve free strikes. Instead, I broaden what each melee action can do after a certain point. Just as someone with a higher Speed can run more per action than someone with a lower Speed, someone with more experience will be able to fire more times, cast more spells, perform more complex psychic actions or deliver a few punches instead of one. Also, the reason why an average civilian only gets two melee actions (assuming they have no martial arts experience) is not because an average person can only punch 2 times in 15 seconds - rather, it's because in the clash and jangle of battle, that's all they'll practically be able to realistically hope to inflict. But if, say, an opponent is tied up, then even an inexperienced person could punch someone 10 times in 15 seconds. In that case, I add more melee attacks as I see fit, or just let logical roleplaying without resorting to battle rules come into play. Another good example of this phenomenon is a sparring match, where someone tries to get more hits than they normally would be able to because it's "fake" and because they want to register a blow. In that case, I let a player choose more melee actions at the cost of each action being less focused: in game terms, add two melee attacks at the cost of -3 to strike or some such.
3. Overhauling magic. Federation of Magic did quite a bit to make magic more balanced, but if you really want viable combat mages, you have to allow magicians to substantially raise their magic spells per melee round. This makes sense, as well: A magician could possibly find ways of speaking faster or controlling energies more efficiently as time went on to cast the spell more rapidly. I also personally add in more healing spells, but that's optional. If you want magicians to play more of a support role, then you can keep roughly to the Palladium system. After all, a tenth level mage tossing lightning bolts using Lightning Arc inflicting 4D6+20 M.D. and having 7 melee attacks is no slouch.
4. Widen the logical type of melee attacks someone can inflict. An elbow, claw, tail strike, headbutt, etc. is going to inflict different damage than a punch. In general, get a feel for what the player is trying to do (say, use their Telemechanics skill to turn off a gun before it fires in lieu of a dodge) and change the melee action cost and the penalties or bonuses.
As I think of more modifications, I shall post them here.