I am just a noob myself but I will give it a try in answering some stuff. . .
Can i only mine my colonies? Do i need to make colony on Venus to place/use automatic mines there?
You can only mine colonies, but in Aurora you can just declare anything (even asteroids) as a colony for free without actaully installing anything on it or having any actual colonists. Even for mining with mining ships you have to declare the asteroid as a 'colony'. Colony is just Auora's term for 'place where I do/store stuff'.
Is it better to build normal or automatic mines?
Automatic mines cost double but they do not require population to run them. So if you have the workforce, regular mines are cheaper for the same result. If you do not (for example in any colony without actual colonists), automatic mines are the answer.
How to place infrastructure/mines that i build on home-world to cargo hold?
Persoanlly, I let the civilians carry all my installations. They seem to have way more cargo ships than I could hope to build. Just make sure you have a nice gfreighter design locked so they can build it and then you can tell them what to move for you by creating supply and demand orders. If you want to use your own freighters, at the task groups window select the task group that has the freighter(s), select the colony that has the installation you want to move and load infrastructure/mines orders will appear. In a similar way you can order the unloading.
Where will i need resources? Only on home-world or on Mars colony too? Or it doesn't matter what colony because they have one pool?
Wherever you plan to build stuff. Every colony has its own resources.
Should i send minerals to Earth from entire system?
If that is where your industrial capacity is concentrated, then yes. If you build stuff in other colonies too, then you need to figure out how you want the minerals distributed amongst them.
Will private sector transport resources for me? Is it automated?
No (but I think I saw something about being a future feature). When you buy the minerals from civilian colonies they sit at the colony they mined them and it is your job to get them from there.
Can i automate my freighters that transport infrastructure to colony?
You can tell them to cycle orders, but then they will start complaining when they run out of infrastructire to transport. I prefer to have the civvies do that for me by using supply/demand orders.
Where can i see how much resources are on specific planet?
Mining tab on the Economy window. As you select each colony on the left you can see the mineral situation on that colony. The stockpile column is the available minerals.
Will mass drivers be enough for transporting minerals? Do i need freighters for that?
A mass driver can send 5000 tonns of minerals a year, which should be enough for the maximum output of most mining colonies, at least at first. You can add more when one is not enough, or suplement with freighters. You only need a single receiving mass driver no matter how many incoming packets you have.
If i don't build anything do i use resources?
If you have docked military ships being maintained, you use a small amount of minerals there. I am not sure if there is anything else I am missing.
Is there something that i should build when i have spare production time?
More factories and labs. Also mines, if you do not have enough but more factories and more labs never hurt.
I create a colony on Mars. Before sending geological survey there. Now GS Ship is scanning Mars but after that (2 times actually) i still don't see any minerals :/
That is normal. Not all planets have minerals. But do send a good geology team and they may find some.
I hope this helps! Also that more experienced players will correct me if I gave wrong advice. Sorry I could not answer the rest of your questions but I don;t know the answers myself.