You can have 5 fleets at most. At the earlier stage, it would be better to have more ships rather more fleets.
4 kinds of ships are available, such as Destroyer, Cruiser, Battleship, Motherhsip that . You can concentrate on using one type, or use combination of different types.
When you unlock a new tier ship, you need to disband fleets consist with prior tier ones and reproduce them to apply the upgrades.
It is recommended to research high tier ships and parts in the same phase. However, if one of them cost too much, you can just proceed another one.