Outside of a source of friendly advice, some beginners assistance, and a pool for stuff (Fleet Bank), Fleets don't serve much of a purpose... despite the Fleet I'm in being at almost 200 members, I only see a half dozen or so on at any particular time and the biggest group I've seen in one place was under 10.
Fleet Squadrons: Fleets carrying out full staffed groups to accomplish all the missions natively. Fleets would get accrued bonuses based on their performance and competitive with other fleets
Competing in Squadron Duty with other Fleets. Special uniforms/decorations/awards (loot drops) accrued to Fleets/Squadrons.
Each Fleet could subdivide into squadrons based on membership (1 squadron per 4-5 captains).
Commodore in command of each squadron
Rear Admirals commanding Flotillas of 2-3 squadrons/Commodores
Vice Admirals commanding 2-3 Rear Admirals/flotillas (minimum of 1 per Fleet)
Fleet Admiral as the commander of a Fleet, commanding Vice Admirals with another
Vice Admiral as "executive officer" and all the abilities of the Fleet Admiral
Fleets can communicate on special "encrypted" channels. Ordinary channels could be listened to by science ships scanning for frequencies, which would enable them to use their sensors to discover and then eavesdrop on other channels. There would be a Fleet/Flotilla/Squadron channels. Fleet Channels would need coordinated science ship action to jam. Non-Fleet "Local" channels would be in the "clear" and "unencrypted" (and subject to jamming from science ships)
Fleets would also get a "headquarters" instances on a space station (the donut/basestar looking thing, not spacedock) that they could congregate at. Promotion/award ceremonies could be held there to distribute awards earned in Fleet actions, as well as elevation in the fleet (commodore to rear to vice admiral).
Many fleet would put 'recruiters' at spacedock to promote themseves, brag about their achievements, etc. The Assembly Room would be used on Spacedock for use of the last winning Fleet to reward them with decorations from <NPC/Dev/bigwig>, as well as send them on special missions, etc.
Fleet Admirals would set their uniform varieties for "dress" and "duty" uniforms. Dress uniforms would be activated in special instances (promotion/award ceremonies in the HQ or Spacedock Assembly Room), but would be optional to Fleet members in "regular" duty.
The mission overhaul (probably due to a separate thread but intimately linked to the balance needed in stuff to do for cruisers vs science ships vs escorts):
Survey missions: Minimal to no combat, mostly scientific
science ships best (easy/fast), followed by cruisers (moderate), followed by escorts (hard/slow)
Report activity for fleet/diplomatic action
-- would need more star systems. 2-3x as many. WHOLE star systems from star to heliosphere
-- unusual spacial phenomena (black holes, wormholes, solar activity, nebulae)
-- unusual terrestrial feature (volcano, atmosphere, etc)
-- identify first contact situations for diplomatic follow-up
-- ECM/ECCM duty (jam hostiles, break through jamming to friendly)
-- AWACS duty (track hostiles from long range sensors) on the move
-- Intel missions (scan foreign planets from long range for duration).
Patrol missions: Intermittent combat with smugglers, odd surprise of battle squadron
cruisers best. Report battle squadrons for fleet action. Report unusual phenomena for science
-- patrolling main worlds and outer systems.
-- diplomatic missions (ferry diplomat from A to

-- science missions (identify unusual phenomena for science follow up)
-- Battle line duty
Battle missions: Current "patrol" system with mostly combat (identified by cruisers and sci).
Escorts best. Report anomolies for cruiser and/or science follow up
-- attacking inner systems, bases, and main worlds
-- escort mission (protect cruiser/sci ship from A to

-- attack duty
Ships would make reports after eqch mission. These would queue up to starbases/contacts for contextual/instanced
follow up. Awards based on successful reporting and coordination to actually clear areas
Science missions: analyze for duration (2-5 minutes). Spend time on interior of ship
for duration. Maintenance and calibration of equipment, shuttles,
samples, etc.
Diplomatic missions: carry ambassador to negotiations (cruiser - native ? NPC for escorts?)
Conversation trees with Ambassador on point A (world, ship, base), on
board-interior, until destination is reached.
Comments
House: Commanded by Klingon Master. Assisted by R'uustai.
Sub-fleet (I want to call it a Fleet but thinking there would be confusion with the Fed term), commanded by a General, who commands 3 Battlegroups
Battlegroup: Commanded by a Brigadier, who commands 3 combat groups
Combatgroup: Commanded by a Commander, who commands 3 captains of individual ships
Your fleet isn't dying because of no Squadrons. Your fleet is dying because your Fleet leaders suck. Squadrons wouldn't have fixed that.