You should try to colonize as many planets as possible, and to grow your existing planets. Build Colony Ships to explore the suns around you. Every sun that you see in the Galaxy Map represents a solar-system that you have not explored yet. Naturally you start exploring and colonizing your direct neighbors. You might find other players pretty soon. You can decide if you want to ally with them, or make them your enemies and try to steal their planets - which might already be quite advanced.
Your planets need to grow as well. You should order your planets to build facilities, in fact always try to keep them busy building something. You can use the Production-Queue to make a planet build an entire list of facilities or ships, one after the other. If a planet does not produce anything, its production is turned into wealth - credits. You should only really do this if you are low on money.
Facilities allow you to improve a planet. For instance, you can build several Farms to increase the production of your Farmers on that planet. Each farm you build increases the food production per farmer by one. So if your planet had a base production of 4 Food, plus you had 3 farms on this planet, every farmer would produce 7 Food per turn!
Keep in mind though that every farm uses up space, and especially on small planets you need to think how to use your available space most efficiently. Farms do also become more expensive to build the more you have of them already. The first farm is the cheapest, but the fifth one on this planet might be quite expensive already.
The more food you produce the quicker a planet also grows population wise. Every excessive food that is not consumed by the planets population, is put into the Food-Storage. Once that is completely filled up it is converted into a new citizen. This is how you grow new citizens.
The same principle is used for Production. The more Production points your Workers produce, the quicker you can build facilities, ships and scans. All the Production points produced on a planet are put into the Production-Storage. Once that is full, the object you are building is available. Excessive production points are used for the next production. No production points will ever get lost.
Of course there are not only farms that you can build, even though in the beginning the list of available facilities is quite limited. What you need to do is to research new technologies, that give you access to new types of facilities, new ship-equipment and other things. The more Scientists you have assigned on your planets, the more Science points are overall produced per turn.
All the Science points produced by your planets combined is used to research a new technology. In the Research Tree you can select which technology you want researched next. You can even click on one further down the tree, which will automatically research all the technologies required to receive the one you picked. Tooltips in the research tree should give you detailed information of what a particular technology does and what it allows you to build.
You need to make wise decisions on how much people you want to assign to Science, how quickly you want to grow your planets, how many people you want to put on colony ships, and so forth. You want to grow and expand, and at the same time not fall behind in technology - because your enemies might be more advanced than you already, which would not be very advantageous when meeting them in a battle...