For modern-era wars, I usually have a single stack of 10-20 drill-promoted armor, and a handful of gunships for pillaging and to counter enemy gunships.
Everything else is air. Use fighters to counter enemy fighters, and bombers to destroy city defenses and for collateral damage.
After conquering a city, I'll airlift more armor into that city while damaged units heal, then I'll airlift 1-2 gunpowder units in to garrison when the armor is ready to move on.
Damaged planes will rest and heal (though it's not always necessary to heal your air units!). Undamaged air units re-stage to the conquered city (when necessary).
Units with 1 move (infantry, SAM, marines, and artillery) are just too slow to keep up with the blitzkrieg, unless they're on boats. Otherwise, they're only good for city garrison duty.
If I've gotten to Mechanized Infantry, the mech infantry can keep up and are also great counters to gunships. The march ability also lets them heal on the move, so they can often beat armor to the front. However, I'm not waiting for Robotics before declaring my first modern-era war. I'm usually doing it as soon as I have a few airports up.
Using this strategy, I can usually conquer a civilization with almost no losses (assuming I have air superiority). It's amazing how fast this happens. Basically, it's one city every 3-4 turns if you don't split your stack; faster if you do.
Without air superiority, I'd expect some losses in the air, and city bombardments can certainly take longer. I might even get a chance to use artillery in this scenario, but it hasn't happened in any of my games yet. I've actually found artillery to be a near-useless unit, but that's usually because I don't tend to fight industrial-era wars without planes.
You can even take out civs on distant continents using this strategy, since you can rely on airlifts to replace your losses once you take your first city. Only problem is, you can only airlift 1 unit in per turn, so if your victim can launch a strong enough counter-attack to repulse you within a couple of turns, your attack is over.
One strategy I've considered is to bring a settler as part of your offensive force. Then you can raze/settle/rush-buy an airport, and poof! Unlimited airlifts on the other side of the world!