A player well on his way to accomplishing this goal.
In the Space Stage, there is no way to "win" the game. It simply does not end, no matter what you have done. First, you complete a bunch of missions from your homeworld, laying down the first few colonies. Then, the stage stretches forward endlessly as your empire expands, your badge collection grows, and your power increases. Eventually, you'll have gained the Omnipotent badge and earned the Staff of Life from the Galactic Core and will start doing whatever you want. But there is only a finite number of things you want to do, and a few hundred thousand in-game years later they will all be done. Once that all happens, you are probably downright bored with that save. One thing to do with your god-like empire is to conquer the galaxy.
Preparation[]
- Create a save starting from Cell Stage. The bonuses you get from Consequences are very important. For Cell Stage, Carnivore is the best option for the additional firepower. For Creature Stage, the Adaptable bonus speed saves a lot of time, Predator 1.5x HP is tempting, but in the long run you won't need much HP after defeating the Grox. None of the Tribal Stage bonuses are particularly important but the discount on the colony tools saves a few trips. The resulting archetype doesn't matter since you can change it later (except to Knight).
- Play the Space Stage normally until you can purchase monoliths. Terraform every purple and pink spice planet near your homeworld to T3 and put a monolith on it. If you're not a Zealot yet, switch to Zealot and use the Fanatical Frenzy superpower to capture the resulting empires you uplifted. This will provide you with easy money whenever you need it.
- Destroy all your neighbors. The way the game works: empires start with one planet and develop as long as you are nearby. Therefore the strongest empires in the galaxy are you, the Grox and your neighbors (assuming you have one save).
- You might want to ally with the Grox to get the Dance With The Devil badge before destroying them, but this is optional.
- Destroy the Grox. There are several methods to do that.
Methods of Conquest[]
There are several ways to conquer the galaxy ("conquer" is stretched a bit here). All of them take a lot of time and persistence, somewhat like normal expansion in Spore, but all are conceptually possible. 
Interstellar Union[]
This method involves consolidating power using a large network of allies. First, remove any Warrior or Zealot empires; these are aggressive and can cause you to get stuck in disputes between allies. Then, simply ally many "friendly" empires such as Ecologist, Trader, or Diplomat, and then spread your alliance network across the galaxy. You can pay your allies to absorb some empires to create a network of relatively few large empires, or you can opt for a more patchwork model and ally every friendly empire you see. This method becomes practically impossible if you ally the Grox (the massive relations penalty necessitates a payment to each allied empire), thus it is better to destroy them first.  Also, you may want to use Monoliths to create more empires. 
War with Everyone[]
On the other hand, you may just want to conquer the galaxy no fuss no mess. This method requires a lot of patience, but it can be satisfying. Essentially, you have to bloat your empire to obscene proportions by colonizing thousands of T0 planets which will not have ecodisasters. This will require a lot of war, and you will sometimes have to wait for enemy empires to expand so you can conquer straggling star systems. Be warned; you will not be popular for using certain abilities to expedite the conquest, such as Fanatical Frenzy. 
Economic Takeover[]
This method is more of a conceptual trick than an actual possibility, but you could theoretically do it. It is simple; convert to the Trader archetype to gain the Cash Infusion ability, be friendly with everyone, and start buying out systems as fast as you can. You will need a robust source of income to continue this strategy, so you will probably have to develop a few Purple Spice planets along the way. Be patient and be prepared for spending billions of Sporebucks buying every system in the galaxy. 
Hybrid Approach[]
This method is a composite of the three methods mentioned above. It is the most practical, because you can entertain yourself and make the most use of your time by balancing between diplomacy, buyouts, and warfare, but you might not get the same sense of satisfaction. Make sure to have plenty of mental strength, as you will be doing a lot of task-switching.
General Concerns[]
Other empires are one of the smallest of your concerns; the biggest time wasters are systems that contain a Tribal Stage or Civilization Stage species. In order to place colony in such a system, you have to destroy every single tribe/city. The easiest way to do this is to de-terraform it to T0, which will instantly destroy the species and enable you to place a colony. The Scientist weapon, Gravitation Wave, may also be used but, due to the large reload time, it's unreliable. Reloading the game is slower than de-terraforming.
Despite popular belief, pirate attacks and Eco Disasters don't increase in number with the size of your empire. Pirates are harmless and can easily be ignored. Eco Disasters can be manipulated to happen close to the part you are focusing on capturing: the game will try to cause an Eco Disaster on a planet without a Bio Protector and Bio Stabilizer, meaning you just have to place one of each on every non-T0 planet except for one which is close to the zone you are about to conquer. This will make you able to respond very fast, therefore you don't have the risk of having to travel half the Galaxy.
Notes[]
- It is very difficult and nearly impossible to complete this goal. The sheer size of the galaxy means it would take a lot of patience and commitment to conquer it. The galaxy in Spore is home to several thousand stars, and most have planets suitable for colonization, by the use of mods or cheats you can make the task of colonizing large portions of the galaxy with relative ease.
- The "Return Ticket" ability might be useful if you wish to go back to your core area rapidly so as to deal with ecodisasters. Consider converting to the Shaman archetype to gain this ability.
- Make sure to save your game a bunch so data is not lost.