Mines are a mere inconvenience...as mentioned earlier, the cost of placing them generally is not worth it for the defender. However, some players do use them...more importantly, the AI uses them. Since high difficulty AIs have more resources than you, that mines aren't cost effective is really a moot point.
There are many ways to get around a minefield, and many have already been mentioned. First and foremost, scout out where you want to send a fleet. You can use a scout ship, preferably with the sensor probe ability (must be researched). Another way is to use the clairvoyance ability on the Revelation Battlecruiser. Since you play Advent, this is a possibility. Though there is much debate on this, I HIGHLY recommend having at least one Revelation for the clairvoyance in any multiplayer game.
The simplest way to defeat mines is, as suggested earlier, to use scouts and anti-strikecraft ships. However, this is slow and only works if the enemy does not have a fleet in the gravity well...something you will only get away with against the AI.
As I think you may have implied earlier, the Illuminators are great against mines...using deceptive illusion, you can wipe out a minefield with your clones and take zero losses. However, this requires activating the ability on each individual ship, since they won't activate it unless an enemy is nearby. Against a human player, this can be too time consuming, though it is extremely successful and can be pulled off if you are fast with a mouse (I recommend unstacking ships in the empire tree). Keep in mind that if the Illuminators are in a fleet, the clones will not automatically join the fleet and tend to stay in front of the real ships, so selecting the clones with click and drag should be easy.
Of course, when you are under pressure to carry out an attack quickly, destroying the mines is likely not an option, especially without Illuminators. The best route is to avoid them. There are two parts to this. First, as mentioned earlier, before you jump, have you ships position themselves outside your gravity well. That way, after they complete the jump, they will be on the edge of the enemy gravity well. When your ships jump in, immediately cancel all orders so they don't move into the mine field by accident. Also, under fleet tactics, tell your ships to "Hold Position" (bottom left button under fleet tactics) so that they don't try to chase after enemy ships (and stray into the mines). The second part is going around the minefield. Simply moving ships back and farther from the planet helps, and then move around the minefield. Be wary, as ships may move FORWARD and turn in an arc instead of pivoting to face another direction. Again, this is slow, and likely not a good option. Your other choice, and by FAR the best, is to move OVER or UNDER the minefield. Using z-axis controls, have your ships go upward, following the edge of the gravity well until they are sufficiently above or below the mines...then move them where they need to be. If the enemy has the mines amongst their structures, fleet, or planet, just make sure your ships stay in their "higher" orbit. You probably want your ships to "Hold Position" under fleet tactics during this.
If you are not familiar with z-axis control, you will need to set a hotkey for it (there is no default). Under hotkeys, find "Global Controls" and set "z-axis control" to whatever you like (I use Z). To move a fleet up or down, click the movement button (green arrow) or press D (the default hotkey for moving). Your ships will now move if you LEFT click instead of the normal RIGHT click while the fleet is selected. To move them in 3 dimensions, hold the z-axis control button and click where you want your ships to go. Yellow lines will show you where you are going and will "shadow" where you will be above/below in reference to the normal plane that battles and construction take place in.
