So if you want to know how the US does it...
The Navy does group ships by type but not in the way you might think. It makes sense to combine a group of DDGs together for certain purposes as they are generally going to have the same equipment, maintenance and training requirements. In the US this is called a DESRON and is commanded by an O6 commodore. The DESRONs primary task is to maintain and train the ships under its charge. However, these DESRON do not work together operationally, at least no more than they do with any other ships colocated in its base. At any one time a DESRON of five destroyers might have one or two units deployed with the rest at various stages of their training and maintenance cycles which makes sense as a single staff can't be evaluating/qualifying all their ships at the pique of the training cycle at once. The only time a DESRON might sort of be in an operational role as the commodore is if he happens to be the senior officer afloat while onboard for whetever reason, but this is really a technicality because while the DESRON is charged with ensuring their ships are trained the Strike Group is the one actually scheduling underway training and is deeply interested in all other aspects of the ship as they are the ones that are going to take it into harms way.
When a Strike Group (or Amphibious Assault Group, or whatever) is formed it is filled out with the various ships among the DESRONs that are at the top of their training cycle. It is the job of the various Type Commanders (SURFLANT/SURFPAC for surface ships, AIRLANT/AIRPAC or SUBLANT/SUBPAC for those units) to coordinate the training and maintenance of all the ships amongst the DESRONs to make sure enough are available for this purpose. The Type commands basically divide the world betweek Atlantic and Pacific to ensure all units to include those based overseas are covered and managed. Working out when units are to be available is a big deal, and since ships are generally designated to a Strike Group over a year before they deploy and the SG commander is usually a one star this can cause some interesting tension between them and DESRON as they have similar responsibilities in that respect but one is senior.
Though the DESRON itself is not an operational unit it does deploy, just not with its ships. Each DESRON trains up an underway staff headed by the commodore that will act as the surface warfare componet commander for a Strike Group. It may have some of its squadrons ships under its control in this role, but not as a the DESRON. Interestingly a deployed staff is still responsible for its squadron at home which leads to a hell of a lot of email and phone call micromanangemet.
CGs and Carriers are O6 commands so they basically operate as a squadron of one, each attached to a Strike Group. There are also independent deploying DDGs and FFGs for various random requirements that will report directly to whatever numbered fleet they are in.
Numbered fleets are the operational commands for the Strike Groups as they deploy depending on which one's geographic area they are in. The numebred fleets report to the COCOMs (EURCOM/CENTCOM/etc.) they are contained within which will have component commands (NAVEUR in the case of EURCOM, the neatness of the system breaks down at this level).
Just a brief peek