It is nearly certain that you'll be declaring war on an enemy civilization during any campaign. You very nearly have to. If you don't attack the weaker civilizations, your can bet that the AI will. The early stages of the game often begin to look like vultures hovering over corpses, as the civilizations with the weakest starts are gobbled up by their militarily stronger foes.
Don't hesitate to get in on the action, but make sure you go in with a plan. Early wars should ideally consist of short, decisive actions against nearby, weak civilizations. You'll want to have as much firepower available as close as possible to the war-front, and you'll want to make sure that you have enough troop transports that you can capitalize on any hole put into enemy defenses. It does you no good to take out enemy ships if you can't take over the planets, because they'll simply build more.
The ideal early war will probably last no more than twenty turns before the enemy surrenders, and you should position your forces so that - upon the declaration of war - you can take several planets within the first five turns. The basic strategy is to load up your main battleships on the future front-line, and then - when you're ready - drive as far into the territory of the enemy as possible. Sometimes they'll catch this and declare war before you reach your target, but most times you'll be able to perform a surprise attack. Focus on the enemy strongholds - mostly worlds with a rating of over 8. The home planet - if in reach - is a must-have prize. Once you decapitate your enemy by taking his best planets, you may not even need to challenge the smaller ones. The influence you gain from the better planets may force the smaller ones into revolt, resulting in a quick, clean victory.
