Reading through all the answers it is apparent that the one thing everyone seems to agree on is that it will mostly depend on the current tech.
If you were to place the battle in early Space Age, where, say, humanity would colonise its solar system and battle each other for assets in the space system, and assume no huge leaps of tech, then the main components of an effective fleet would be:
The main focuspiece of any fleet would be its carrier. The carrier would host and deploy its fighters and most drones.
The Drones themselves would form a type of 'sphere' around the carrier and they would be mostly tasked with intercepting missiles, and cause havoc to fighters. They would be armed with thin lasers, that pre-detonate any missiles. They wouldn't be as effective against agile fighters, and assuming some armour to them, would not cause that must damage anyway.
Frigates would be mostly armed with conventional balistic weapons and have two uses: close quarters brawlers, flinging small bits of metal or shells at enemy ships, that are hard or impposible to intercept, and, at larger distances, would aid the 'missile defence grid' and act as better decoys, trying to draw some fire from the carrier. They would also be the best defence against fighters, given their close range capabilities (think minigun-armed hedgehogs).
Frigates would also encircle the carrier, or be arranged in a sort of 'wall', with the side facing the enemy. They would also host their own drones.
Fighters, most obviously, would be the main attack force. They would have to be numerous and agile, but small enough so that they are difficult to detect. They would be tasked with reliably getting through the wall of drones and planting missiles in the enemy carrier (or frigates). However, they would mostly be stealth bombers. They could have something to use against drones, such as miniguns, but speed would be their main asset.
Fighter dogfights would probably only errupt in deep space, if the squadrons met each other before reaching the enemy fleets. However they would either last too long or too not long at all, given the extreme speed and range. My guess would be that engaging in dogfights near fleets equipped with these armies of drones would be unfeasible for the attacking party, and it would retreat and retry after it out-manevoured its enemy.
Smart admirals could potentially make use of large gravitational masses, such as Jupiter, to fling projectiles around, basically making certain areas a 'shooting gallery', and then try to manouver his/her enemy in those galleries.
Asteroid fields would be perfect ambush locations, assuming ships can lower their power output and float around the fields like rocks.
I'm also guessing that how the fleet is positioned in regards to the sun/star of the system would also affect targeting/spoting/systems.
Other space anomalies, such as some glass clouds/nebulas might also create hazards that are either impassible or highly dangerous, but can also be used as hiding spots/ways to lose enemies.
Another idea would be to use frigates to do high-speed drive-by's at effective cannon range from the enemy fleet, however trying to drive a wedge through the enemy fleet would probably be a bad idea, since your whole fleet would have limited evasion capabilities, while being subjected to the full firepower of the enemy broadsides. Also, the lack of active drones would leave it vulnerable to nukes.
In any case, the way I imagine space fleet battles would work at the momment would be two blobs trying to shoot at each other from large distances, trying to find weak spots in the enemy's drone shield. Flanking would work if the enemy has their drones arranged in a directional cone, in order to better soak up damage from a known fleet.