Master of Orion is one of the iconic 4X games. eXplore, eXpand, eXploit, eXterminate.
This does not mean that these phases of play are mutually exclusive - some degree of overlap is inevitable - but this is characteristic of a territory-claiming game.
eXplore[edit]
When you first start the game, you have two scout ships and a colony ship and no knowledge of the surrounding stars. Your first priority will be to explore the stars that are within 3 parsecs of you, and send your colony ship to one of them.
It is often best to send your colony ship off to help explore these stars - send it preferentially to a yellow star, or if none are close enough a green star, then a red, then a white; blue and purple are rarely habitable with starting technologies. Stars in nebulae are more likely to be rich, which is a good tie-breaker; stars lower down the map will be reported on after stars higher up the map, making it easier to re-route your colony ship if the scout ships have found somewhere better.
eXploring the galaxy depends both on developing longer ranged ships, and also on early eXpansion; by placing colonies on new stars you often enable your ships to reach still more stars.
During this phase you probably should research improved eco restoration after you have built 80 factories on your home planet, if it is available. Improved eco restoration reduces pollution and significantly reduces the cost of constructing colony ships.
eXpand[edit]
This phase of the game is usually most evident in the early to middle stages. As soon as your home planet is fully kitted out with factories, it is often desirable to start building colony ships continuously (unless you need range 4 or range 5 to reach anything worth colonising, in which case you'll have to divert some resources to Propulsion research.)
Rapid development of your colonies is important here; you'll want to send at least a few colonists from established planets to new ones whenever the new ones are settled. Secondary colonies do not normally do anything more than build factories until they hit MAX, after which most of them switch to research, possibly with a little diverted to defense and to population growth. Rich planets will hit MAX sooner and then you can build colony ships there.
Researching planetary technologies is key for the expand phase - the more hostile the planets you can colonise, the better! Once your territory stretches across more than 10 parsecs, higher speed for your ships (and colonist transports) becomes more and more desirable and you will probably need to research new Drives in your Propulsion techs.
This phase is fundamentally a land grab (OK, a star grab!) - you want to claim as much territory and as many nice planets as you can before you come into significant conflict with the AI. Hostile planets are useful stepping stones and also useful territory in their own right because the AI often can't invade them.
eXploit[edit]
Of course you are always trying to wring the most possible resources out of your planets - right from turn 1 you are building factories on your home planet so that you can exploit its resources better.
Still, there is often a distinct eXploit phase, after you have colonised all the vacant planets you're going to get to, and before you begin the wholesale destruction of your opponents. This is the time in which you research all fields of technology simultaneously. This is the time in which all your planets build up to a full complement of people and factories, and maybe a basic minimum of defenses; then they get down to some serious research, and often then into investing to enjoy the fruits of that research. If you are not engaged in any conflict during this phase then the turns can pass very quickly.
Key techs for this phase are Robotic Controls and Terraforming, both of which increase the amount each of your planets produce. Industrial Tech will make it faster to build up to that point; Reduced Industrial Waste and better Eco Restoration will mean there's less resources lost to waste. (If you have high Eco Restoration then Reduced Waste has little impact.)
You usually also need some defensive techs to keep the AI off your planets while you eXploit them. Force Fields are useful here - normal shields and Planetary shields; also Personal shields to help fight off the troops that do land. Construction yields both armor types and suits that help your ground troops - the armor is particularly useful since it will also make your missile bases and ships tougher. Weapons tech will let you research rockets or missiles that mean your missile bases do more damage, and hand weapons for your ground troops. Space Scanners (computing) will help you anticipate the arrival of attackers, meaning you have time to build up defenses or send a few ships before the enemy arrives; this often means the difference between a painful blow and a successful repulsion.
For this phase to work out well for you, you really want to be able to do more research than your opponents. This usually means having more planets than your largest opponent, and if this is a problem for you then after a minimum of eXploitation you'll be well advised to crush the weakest AI and eXploit their planets.
Sometimes the eXploitation phase is not so obvious because you *are* engaged in conflict with one or more AI. In order to get ahead overall, you don't want to divert too much of your empire's investment into war. If you have a tough struggle with one AI opponent and you haven't been doing any research, everyone else will have a huge advantage over you. Usually building ships only on Rich planets will work out well, but this is critically dependent upon how many you have.
eXterminate[edit]
In order to win the game you usually need to kill your opponents ships and take over their planets. If you've built up enough of a technological advantage, this may be as straightforward as building up a big fleet and using it to smash everything in your path; it is quite possible to steamroller your opponents into eXtinction in 20-40 turns.
If the game is challenging then you will probably move more gradually into the eXterminate phase, starting with the weakest opponent and adding their territory to your own. This can start as early as researching class IV Deflector Shields, as these will completely stop Nuclear Missiles and Lasers; if your opponent has not moved beyond those, then ships with shields IV and nuclear bombs can clear the paths for your ground troops to invade.
(If you're right close to the opponents you can win some planets very early on, before anyone has any defenses - this is a simple matter of sending troops to their planets, and the Sakkra and Bulrathi should be particularly good at it.)
Offensive techs are key in this phase. You want some sort of Bomb - Fusion Bombs will be sufficient until quite late in the game, but bigger bombs are of course better. You want a nice fat beam weapon - preferably something more penetrating than Ion Cannon, but this depends on where your opponents are up to. Early on these may go on different ships, but in the later game you can build ships with both.
You also want your ships to be as tough as you can make them; high armor (Construction) and high shields (Force Fields).
Ships with High Energy Focus are particularly effective - that is one of the very few techs in the game that you are guaranteed to get access to; it's in the Propulsion stream. Of course high speed drives for your ships are also valuable, almost essential.
Planetary techs also help, especially cloning, as they reduce the cost of rebuilding your population; you will want to divert resources to Ecology on any planet where you send away large numbers of transports.
Everything that made your ground troops stronger in the eXploit phase is still valuable here. Your troops' effectiveness may grow in great bounds if you missed early ground-combat techs and then research later ones in the same sequence.
You will be continuously building ships (at least on rich planets) during this phase, and it is often common to route those ships to newly conquered planets; keep an eye on enemy ship movements so that you can bring the main fleet back if their main fleet is headed for a place you only have lesser forces. (If this fleet includes particularly slow ships, you may be able to get around them and then get back again - or even conquer the enemy so quickly that you never have to face the main fleet!)
And eXterminate the opponents, one by one. Once you have wiped them off all their planets, their ships will disappear and they will harry you no more - usually meaning your forces are free to move on to the next opponent.