First thing you need to do when starting the game is go to the designer and create light versions of your colony and constructor ships. No need for extended range or increased speed for close planets and resources, just wastes build time. Also design a light defense type craft, revisit as you research and pay attention to construction cost. You dont have much manufacturing at the moment so dont add anything else. One weapon, thats it, no engine, no range, no sensors, no defense. Dock all military ships on your planets and starbases to lessen maintenance costs. No need for them to be sitting in space draining your economy. While you are in the designer put as many sensors as you can on a cargo design to act as your sensor probes to place in a 360 around your colonies and/or on each colony as needed, use them to clear the fog of war and explore.
As you gain more tech and manufacturing update your ship designs or create knew ones to give you the most weapons for the least build time. Make sure you have light designs for shipyards that have less manufacturing ability in your empire. If you are building a ship with 27 turn build time, that is the wrong answer and a waste of time. Make a lighter design or build something else.
After you have queued a shipyard in a newly colonized planet add that planet to an existing shipyard until it is built. Even if it is far away, you will get one point minimum which is big early game.
Build a couple of over lapping military starbases and keep most of your fleet within their overlapping coverage to increase your military power. Preferably near your homeworld or strategic area you have determined.
If you play on a big map it may be in your interest not to explore too far past where you intend to colonize and keep. If an alien race hasn't been discovered yet, they are not going to declare war on you.
Trade your early tech for money with other civs or for tech you need for your strategy.
Tip. If you hold the control button down while a ship is selected the number of moves and turns to a hex will display as a tooltip. Great for knowing how much range to add to a ship.