What do I research?

Depends on what aspect of the ship you want to make better.
Hulls for larger and stronger ships.
Weapon techs for more powerful weapons, ditto for defenses.
Logistics to increase fleet sizes.
Propulsion techs to improve speed (number of movement points per turn)
Sensors to increase how much fog of war is uncovered.

Then, design your own ships, don't use the core ships that come with the game.

First some drives. Good to get up to Impulse.
Second basic Trade.
Thirds start researching governement and good and evil.
Some other weird tech to research are Planetary Invasion and Planetary defenses. It is a real pain to get them from AI.
Finally Xeno Farms.

DO NOT RESEARCh weapons/defenses, minutiarisation and logistics. AI will do it for you,Just exchange. Also note AI will probably get you better Factories/Research labs then you could research so wait for these ones too.

Note: this is good for patch 1.3.0. Dont know how patch 1.3.1 will affect this.

Thanks I have been getting slaughtered by the AI and I am on Beginner.

What do I have to research to build better ships? All I have been able to build is Star Furys. What do I have to research?

It sounds like you're trying to build core ships as they become available. You really need to pretty much ignore the core ships (I just obsolete each and every one of them as they become available). In all cases, you're much better off designing your own ships based on the best tech that you have at the time. If you're new to the game it can seem a bit daunting at first, but it's actually easy and fun. Plus you will continue being slaughtered if you rely on core ship designs.

Core ships are horrible to motivate you to use the ship designer which is the way the game is intended to be played.

For me, I research engines up to Impulse Mark II and Missles up to Stinger III right away. I tend to trade for defenses and life support since they're usually pretty cheap.