Welcome to the game and the forums.
As for your questions:
1) To destroy all ships or not before destroying the Command Station (cap ship).
This can go either way. Enemy ships have two states, "Guard" and "Loose". Ships that are Guarding are tied to something, usually a guard post or the command station. Ships that are Loose are considered "Threat" and show in the Threat counter at the top center of the screen.
Ships that are Threat (usually just called Threat) move towards your empire and attack when they feel it is a good time to. They do take into account how powerful your forces are on the other side of the wormhole before attacking so they may attack or may wait on the AI side of the wormhole for more ships before attacking.
When you destroy the Command station, all enemy ships in the system loose themselves and become Threat. (If there are still guardposts alive the ships in Guard mode at those post may turn into Threat or may stay in Guard mode.) So you have do make a judgement call, are your defences in your own system strong enough to hold against all the enemy ships left in the system turning into Threat and attacking your system?
2) Ships moving slow.
There are two things that slow ships down, Engine Damage and Gravity Effects.
If you mouse-over a ship moving slow you should see in yellow a "Speed Reduced to: xx" number that indicates a ship is being slowed. If the engine health is not at full it is engine damage and the ship needs repairs to move at full speed.
However, it sounds like you are running into a Gravity effect such as from a Gravity Turret so you have to find the unit emiting the gravity effect and destroy it.
Note that your own Command Stations give a speed bonus to ships in the same system so ships in an enemy system will move slower then in your own system as a general rule.
3) There are two types of guard posts that make the Command Station invulnerable and have to be destroyed before the Command Station can be attacked. Command Station Shield Guard Posts and Counter Attack Guard posts are the two.
If you mouse-over the Command Station it will have an "External Invicibility: 1x" line showing how many Guard Posts are providing invulnerability to the Command Station.
Note that the Warp Gate does not get the invulnerability, you can destroy a system's warp gate while the Command Station is invulnerable.
This will vary from player to player depending on how things go.
My personal opinion is Bombers Mk II and Fighters Mk II are a good starting point. The Harvester Upgrades never hurt as they give more resources but they don't increase your combat power so that is a personal preference you'll have to decide on.
Hope that helps.