Researching new technologies is easy. You just spend money on Tech in the Player Window, and away you go. You can weight your research into certain areas by balancing your Technology budget in the Tech Spending Window, available from the " Windows " menu.
Range is how far a fleet can move before refueling. If a ship has a range of 8, it can go 4 spaces and come back to refuel, or it can go 8 spaces and be stranded when it gets there. When a ship's Range is 12, it can go 6 spaces and come back safely, or make a one way trip of up to 12 spaces. That's a long way, and it will let you make deep raids into enemy territory or explore planets on the fringes of the Galaxy.
Speed is how fast a ship goes. If a ship has speed 1, it will take it five turns to move five spaces. If its speed is 3, it will make the same trip in only two turns. Speed can be important for reacting quickly to an enemy attack, or surprising him with one of your own. A high speed technology also can allow you to explore new systems much faster than you would otherwise be able. Perhaps most importantly, in battle, the ships with the highest speed shoot first.
Weapon and Shield technologies determine how well a ship fights. The amount of damage you do is based upon the difference between your Weapon Tech and the defender's Shield Tech. If your Weapon Tech is higher than the enemy's Shield Tech, you'll do a lot of damage each turn to his ship. If your Weapon Tech is less than the enemy's Shield Tech, you'll do little or no damage each turn.
For example, let's say your ship has Weapon Tech 5. If you attack somebody who has Shield Tech 1, his shields will be ineffective against your weapons, and you'll decimate him. If the defender has Shield Tech 5, you'll damage his ship somewhat, but his shield will block the rest. If he has Shield Tech 8, your attack will be almost 100% blocked, and he'll laugh at your puny attack.
Generally, to fight an even battle against an enemy with high weapon and shield techs, you need twice as many ships for each level you are behind. For example, if you have Weapon and Shield Tech 5 and the enemy has Weapon and Shield Tech 7, you want four times as many ships for an even battle. In fact, you probably want more like 10; who wants an even battle?
Mini (miniaturization) allows you to build ships for less metal but more money. Since metal can be a lot scarcer than money, building ships with high Mini Techs may be a good idea. Colonists can't be miniaturized, so Mini has little effect on Colony Ships.
Radical research can give you things that none of the other technologies can. When you put money into Radical Tech, every once in a while your scientists may make an amazing discovery. Some Radical discoveries might give you higher tech levels, biological weapons, weather control, mind control, cloaking, or new mining techniques. Whatever.
You never know when the next Radical Tech leap will happen -- it's far more random than the other technologies. We're not going to describe all the possible things that can happen with Radical Tech -- you'll have to discover that for yourself.
Radical Tech increases to the other technologies don't change your baseline of research for those technologies. For example, if your Weapon and Shield Tech levels are 5 and 5, and your Radical Tech causes your Shields to go to 7, your Weapon Tech will reach 8 at the same time that Shield Tech does. This means that the Radical Tech advantage is only temporary, so use it well.
The technologies you choose to develop are totally up to you, and will depend on your goals for that game.
If you develop a high Range Tech, you can reach many planets on the fringes of the galaxy that nobody else can reach. You can develop these without having to worry about defending them.
Good weapons and shields are essential if you're involved in combat, but remember, the technology alone isn't enough. Having the capability of building high Weapon Tech spaceships means nothing in combat; having lots of high Weapon Tech spaceships does.
Miniaturization becomes more important as the game goes on. In the early game, there is plenty of metal in the galaxy, and it's easy to come by. Late in the game, it might be the limiting factor of your fleets' sizes, and a high Mini will serve you well.
Speed has advantages that aren't immediately apparent. A fast counterstrike fleet can often catch and destroy an attacking enemy force after it has captured one of your planets. Usually the attacker has to fly in a Colony Ship, colonize your ex-planet, and support it for a turn just so he can refuel. If you have fast ships, you can catch him there. If you have slow ships, he'll get away unscathed.
Radical Tech is a gamble. If you happen to get lucky, and make useful discoveries quickly, it can turn your game around. On the other hand, sometimes you'll get nothing out of it for eons.