Darkadder put a pretty good post up on a strategy that uses marines... the only continent invading strategy I've seen that uses this useful unit.
Assuming you don't have huge stacks of carriers, bombers, and Modern Armor...this is a strategy I have successfully used to invade the AI continent in the late industrial or modern era. To keep it simple, I'll discuss these ideas conceptually, rather than giving lots of hard numbers.
The strategy is designed to seize and maintain the initiative from the opening round of the war, which *I* initiate, usually with that first attack from the sea. The basic idea is to hit the AI fast, and hard, in that opening round. (Darkadder and I agree here...)
These are the basic goals of these tactics:
1) Take, and hold, a coastal city in the AI territory, on that first turn.
2) Take, and hold, at least one inland city in the AI territory, again, on that first turn.
3) On the second turn, move significant quantities of follow-on forces, to maintain the initiative.
There are 3 basic ground units (or unit types) I use:
1) Marines
2) Good defensive units (Infantry or Mech)
3) Fast offensive units (Modern Armor, Tanks, even Cavalry)
While you can play with the numbers or augment this strategy, the absolute MINIUMUM invasion force would be one transport each of Marines, Infantry, and Tanks. If you use Cavalry, load 2-3 transports of cav to substitue for 1 transport of tanks. The choice between tanks versus cav is one of range and speed for hitting that second, interior city (unless you've got Modern Armor or Panzers). The more transports of fast units, the better. Of course, these basic ground units may be augmented by artillery, if you like - but it's not fundamental to the strategy.
The most important naval unit for this strategy is the transport. You almost can't have too many of these. I'll use enough Destroyers and/or Battleships to cover the transports. While an aircraft carrier or two with bombers can be used to "soften up" the initial invasion point, there's a downside to this that I'll discuss shortly.
This strategy requires a bit of forethought, planning, and prepositioning of forces. This includes the following:
1) Identify that first target city. This will be a coastal city, preferably a smaller one, with a harbor. The harbor is important!
2) Identify an interior city within striking distance of this coastal city. By "striking distance", assume you have a already have that stack of fast attack units stationed in the coastal city; you should be able to attack the interior city IN THE SAME TURN. Assume a 9-square radius around the coastal city. If you will have to cross 2 enemy squares to hit that city, you'll need lots of Cav or Modern Armor (or Panzers or Cossacks). If there's a city that you can hit after crossing only one enemy square, it's a great target for Tanks!
3) Position 1-3 transports within range of that first coastal city - even before you can build Marines or Tanks... By "within range", assume you already own that coastal city, and these transports are full. You should be able to move the transports into the city and UNLOAD them IN the city (in one turn). This position must be outside the cultural boundaries of the target.
4) Position "convoys" of transports from one of your ports to these forward transports. The idea is to be able to move one "boatload" of ground units to the forward position in a single turn. (The mechanics of the "one turn" trans-oceanic move should be discussed elsewhere on this site). If you can build a 2-3 deep convoy, that's fine, but the basic idea is to be able to move one transport load of follow-on forces across the ocean in a single turn, and to do it each turn until you can't fill a boat at the back end.
Okay, we've done our planning and reconnaissance, pre-positioned our forces. For whatever reason, it's time to wage war against that AI across the sea. It's time to...
SEND IN THE MARINES!
First, I move the transport(s) containing Marines to the square adjacent to the coastal city. I wake 'em up, and attack from the sea. Usually the marine that kills the last defender will be down to a hit point or two... I'll then move the transport into the city, unload a fresh marine or two, and fortify them IN the city.
Next I grab the transport full of defenders (Infantry or Mech), move that transport into the city, and UNLOAD it. I then position (and fortify) the defenders in the squares NEXT to the city, not within it. My goal here is to form a perimeter AROUND the city I've just taken. An artillery piece under each infantryman isn't bad, and 2 infantrymen and an artillery piece make a pretty tough perimeter. This gives me a good defense for the city, and a pre-positioned counterattack in case of a culture flip. The 2 or 3 marines I've stationed in the city provide a little more than a mere token defense in case the enemy counterattack breaks through my perimeter. These 3 units will also quell resistance in a few turns, allowing me to "buy" some culture (Temple and Cathedral) to prevent a culture flip.
For the final phase of the initial attack, I move the fast attack units into the coastal city, and UNLOAD them, with all their movement points intact. I then attack, and take, that inland city. If I can get some additional defenders in there after I take it, I will. I will try to leave one or two decent attack units along the perimeter of that coastal city, in case of a culture flip. If you've got enough fast units to take more cities, then by all means do so...
The AI's next turn will determine how well you've planned and executed your invasion, and whether or not you've underestimated your enemy. He/she will throw most of what they've got at you in the counterattack that will ensue. If the coastal city culture flips, you'll face one regular infantry/rifleman, plus maybe a conscript or two at most. If the AI recaptures the city militarily, you'll probably face more than that... If you can hold that coastal city for the next turn, you've bloodied their noses pretty good. If you hold both cities for the next turn, you've rocked him back on his heels a good one. If you hold both and take another city on your next turn, and hold on to these cities through the second turn of the war, then the outcome of the whole war is probably decided.
My convoys of transports will allow me to move and unload one full transport of Tanks or Modern Armor into his territory for the next 3-5 turns. By then I should have reached the turning point in the war. While using an airport has its merits, you have to wait until the next turn to actually USE the units (and risk losing them in a culture flip.)
These are my thoughts on invading an AI continent. Additional commentary is below. I welcome feedback on this.
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A note on bombardments:
While I'm sure there are those on the board who would advocate using carriers and bombers to pummel that first city down to a population of one, I prefer not to, for the following reasons:
1) All of the city improvements will usually be destroyed in the process. While you won't capture a city with its cultural improvements intact, if you capture a city with a harbor and a marketplace intact, you'll have a LOT less trouble with civil disorder after you take it - especially if it was a smaller city to begin with.
2) If you capture the harbor, any new resources or luxuries you capture through the course of the war will be "linked" to the rest of your empire. This can be critical if you've just liberated some aluminum you need to build the spaceship...
3) A carrier full of bombers represents 580 shields, 20 shields less than 6 marines. Six marines will likely do more damage to the enemy UNITS than four bombers. If you use this strategy, your marines will suffer casualties. This seems to be the nature of Marines, deal with it. If the city appears big and well defended, use a second transport of Marines.
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Notes on my warfighting style:
I prefer fast units, and a blitzing offensive style. In ancient times I prefer horsement to swordsmen.
I rarely pillage or bombard. I prefer to capture cities and territory intact, and I'd rather keep coming at the enemy with horsemen or cav that will live to fight another day than continue to feed slow units to a meatgrinder.
I like short, decisive wars. My personal record (on a pangea map) is 13 cities in a single turn, and the remaining 3 cities the next. I did this with a combination of Modern Armor, Mech Infantry, and Cavalry, using ancient age units to police up AI workers along freshly liberated railroads.
) If they aregetting an oil source from them then destroy the people they're getting it from:die!: 
Keep in mind remeber to use stealth bombers otherwise this idea will be a waste of time to read
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