First you have to research Planetary Invasion.

After that is done a new type of module called the Troop Module will be available to you. This troop module carries 500 million troops.

You will also get a new basic ship desing, build one of those ships, or make your own design if you want to make it go farther, faster, carry more troops ect.

Then once it is built, launch it, make sure you have enough population for at least 1 billion soldiers. In my experience 1 billion soldiers can invade a planet of about 6 billion.

Now make sure the planet you are going to invade has no ships in orbit. If it does destroy them, then just select your Troop Ship and click on the planet. The troop ship will fly to the planet and land the troops. An invasion screen will pop up, if you have further Invasion techs you can choose to weaken enemy troops before invading. If not then just click on Traditional battle.

A screen will now be in front of you, there will be numbers quickly changing near the middle of it, this will represent the odds you have of winning and the odds they have of winning, try to wait for a nice high number on your side and press beggin battle. Watch the (in my opinion ugly) battle or chose to skip. If your troops are still alive then you get a new planet.

Good luck in your conquests