How do you cope with the Fog of War?

     Research sensors early and rush buy Eyes of the Universe on the longest lease (also Nano Recorders at level 3 - cheap to buy or build).    Eyes of the Universe pays for itself because it helps you find minor civilization and resources.

    Make several extra survey ships, I wouldn't bother with scouts. 

A few survey ships won't do the job on gigantic and immense. I want to see what the AI's are up to.

How quickly do you actually build the eyes of the universe?

I try to have it 1 to 1.5 years in.

In my current 9 AI/tough/60 planet/huge map game (my favorite kind) , I over-expanded and did not meet any AI to mooch cash off. So its taken me almost 2 years to build the Eyes. I also have about 50 Scouts I got from the enemy and 4 Research Vessels. It looks ridiculous and overcrowded. I"m shaping it into a grid, but that's not much of an improvement.

I'm not too experienced with the game, though. Maybe it's better to go mostly blind.

  Sensors on starbases usually do a lot for me.

  I put sensors on my space miners then position them when they are not building on asteroids;  and I have sensors on constructors which wait near my resources ready to make optimal starbase-weapons upgrades.

With tech trading/brokering on, I research to Star Democracy, and then switch to sensors.  At level 3 I build or buy the nano-recorders, and at level 4 I build the Eyes of the Universe (usually on the longest term lease). 

I put sensors on my freighters. Typically, I'll already have high-level sensors by the time I have one of the "advanced" trade techs researched, so I usually develop a "sensor freighter" design. It's good for keeping an eye on the AIs especially when combined with Eyes of the Universe.

Actually I've never done much about it. When colonizing I send my ships to clusters of planets. I may do some extra flying around a sun to see all of the planets befor choosing the best one. But, yes, there have been times that I later find out that i JUST missed something nice.

Early in the game, you can simply click on the star, and it will tell you how many planets are habitable and already colonized.  What you will often miss will be resource nodes.  That is why researching sensors early and building eyes of the universe can be helpful.

  This sort of thing occurs a lot ,  with my survey ships on auto-survey (I put engines or life support on them instead of sensors,  early in the game):

This is true whether or not you have Eyes of the Universe. The only difference between having Eyes of the Universe and not is the distance your ships can see.

If you turn off the color display of each AI's area of influence in the mini map then you can see the areas where you've cleared out the fog of war. Areas that have not been cleared will appear gray whereas cleared areas will appear black. If you do this and have ships displayed on the minimap then you can *always* see ships as long as you cleared the fog of war from the area at least once. So while you won't see new ships move into a previously cleared area on the main screen unless you have ships in range to view them as long as the fog of war has been cleared you can find all enemy ships via the minimap. You may have to zoom in somewhat and scan the minimap thoroughly to find them and it may be difficult to tell Iconian ships (white) from Korx ships (light gray) but you can see all ships that way. At least with DL v1.53 and DA v2.01, not sure about TOA.

Happens to everyone, I think.  It's hard to beat the AI when it comes to finding and grabbing resource nodes.  Check the thread on Index of Tricks for ways to get that node for yourself.