Dancing Lessons
No matter whether you take an aggressive or a conservative attitude toward attacking your enemy, one of the keys to winning is patience. A simple bull-rush to take out the enemy in one strong push can sometimes work, but just as often your fleets will be too evenly matched and your ships will be destroyed. Instead, try a little finesse, use the map and learn to "dance."
Dancing is the art of moving your fleet from planet to planet, avoiding pitched battle with the enemy fleet and working your way to an open nuke on an unprotected planet, an opportunity to explore a new jump, or other strategic move. You can often reinforce your fleet as the dance continues, moving in ships from a nearby builder or stargate jump point.
While you can use this tactic with your entire fleet, the most effective form of dancing divides your shipst, forcing your enemy to defend more planets than he otherwise would. He also has to maintain more ships to cover all of your possible attacks, forcing his tech development to be lower than yours. The grand finale occurs when you manage to bring your separate fleets back together into one. With luck you will be able to set one fleet to nuke a planet and bring the other one in to support the nuke. If, on the other hand, your enemy can match your fleet in that planet in that turn, you simply redivide your fleet again and start over.
Another time when you can begin the dance is after a pitched battle where several of your ships survived but at low BR. In this case you are perhaps in a worse position than your opponent. He can rebuild with full resources, but you have to support all those damaged ships during a rebuild even if you do set them to dismantle. Unless they are very low in BR, though, it might be more advantageous to start them dancing. Your opponent has to respect them and cover them as if they will return to full BR, so if you can keep them moving for several turns you can return them to full BR without having to spend the resources of rebuilding your fleet.
When used against any but the most experienced players, dancing can be a form of psychological warfare. Having to defend against multiple fleets can drive any player crazy, and most simply cannot cope with it. When you attack with a single fleet you have fewer options, thus making the defense easier. This strategy does require time to learn, but is well worth the effort.
Stormer's Tactics & Maneuvers and Gooseberry's Stellar Crisis School are both excellent and specific in-game examples of how to use these principles in several different ways.