There are already a lot of indications from your post to answer many of the below, however, for future readers I would expand the answer and then go into details with your specific case. Also, a lot of the previous answers apply, therefore, while I will be repeating them, that would be intentional to slot them in a wider concept.
I am wondering how you move your fleets during combat.
First we need to know if you are attacking or defending. Generally speaking, to split your fleet is never a good idea, unless you have designed properly ships with defensive duties and other more for offense/fast attack. The rule number one to remember is that fleets move at the speed of the slowest ship included. So, if you have some commercial support ships in you fleet that move at 2,000kms while your fast frigates could close in the enemy at 6,000kms, you won't be able to do so unless you detach your slower ships first.
Now, splitting your fleet is suddenly not a very bad idea especially if you are on the offense! If you are defending, then a lot of the movement will be determined by the battle landscape. Are you defending in your home system, at a strategically valuable location, empty space?
When defending, I would try to stick closed to STOs (if any) just to add some extra firepower. I would probably keep the fleet together as well and take advantage of the full AMM/PD force to counter incoming enemy salvo. Again, it really depends on how you have structure your fleet, and the situation.
Another very important thing that would determine how I move is the intel I have on the enemy. When I am expected to be in their sensor range, firing range, speed, and so on.
I do not even know how to set fleet speed. I pressed set speed button but did not see any changes of speed.
I have answered the speed question partially before, however, to ensure you are travelling at a predetermined speed, you also need to untick the use max speed checkbox. By doing that you are able to set your preferred speed which cannot exceed your maximum speed that, as said, is the maximum speed of the slowest ship in the fleet.
Even more interesting is it possible to make something like keep specific range command if i want to kite or pursue the enemy?
Yes, you can. As it was said previously by nuclear, you need to flag the contacts checkbox, double click on the contact displayed or click on follow then add move. To keep a specific distance, you must first set of which distance you want to be then click on the order. If you did everything right, you will see the order follow along with the distance you set. Later, you may start also exploring formations, which allow you to keep ships from the same fleet distanced at a certain bearing. I find that useful when I am in search and destroy missions and need to cover larger areas with smaller sensors.
Do you just use micro manage with navigation points and manual speed control?
I use waypoints a lot and would generally use speed manual as I find the follow order to don't suit my needs especially if you lose contact for a bit with the target for whatever reason. This is true only if I do not have a clear advantage on the target though. So I would say that the easiest the combat is expected to be the less management I put in the fight.
If my fleet lost enemy contact can I maintain bearing and speed of me fleet?
Same as previous answers from nuclear and I, I prefer the use of Waypoints so that I am not in need of adjust my bearing. Eventually, I widen my search by using formations while keep moving towards the waypoint at the same bearing.
Is it possible to keep maneuvering around low speed target keeping my speed at maximum to minimize enemy hit chances on me?
Your to hit chance either in hit or being hit it's affected by the single ship. Meaning that all the limitation in speed by having slower ships in the fleet or even defending stationary on top of your planet, would not impact on the calculations. While it is not realistic, Stave reckons that it is the easiest way to deal with the problem of coding every single ship possible % at the time of combat. This wouldn't even consider what if you change speed in between? It is also a matter of calculations that would be required especially on large battles. Which is also that same reason why fuel consumption is not determined by distance covered instead of speed.
Long story short, the only reason why I would circle around a slower target is to avoid accidentally end up in their firing range, but it makes no difference "combat wise".
PS. I am talking currently about close combat with lasers. I did not use missile yet.
FC set? Weapons assigned? AS on? Shield on? Contact in range? Sync Fire or just Fire all?
Then, good luck sir, put up a good fight and die with honour. No worries, we all been there and as a matter of fact, many of us still geting there more often than we'd like to admit.
