Three planets (asteroids [other than the dead variety] are typically counted as planets) is a very tiny empire.
On maps with few planets, you really do have to pinch every penny. In these circumstances, you really need to avoid civic technologies like the plague. Civic technologies can take upwards of 20 or 30 minutes to pay off for a large empire, and for a small one could take hours. Pick technologies very sparingly and for tactical reasons. In these situations, the only reason to get civic labs is so you can colonize ice or volcanic planets and then use them as military staging points.
If the map has more planets available, you really need to expand faster. Capturing a new planet is way way way more profitable than putting up trade ports or buying economic technologies. You should buy enough population upgrades to get out of underdevelopment (one upgrade for asteroids, two upgrades for all other types of planets), and also build all your extractors, but early game avoid trade ports and refineries. Sometimes culture is worthwhile early on, but even then that's not a big deal.
For defense, use a combination of repair bays and medium tech-level units. The long range frigate (Javelis, Assailant, Illuminator) makes a great backbone for any fleet, and if you combine it with the flak frigate (Garda, Sentinel, Defense Vessel) the AI won't be able to dislodge you with early game units alone.