In the laning phase it's all about wave control. I usually want to push the first wave (but not too hard) and pop vitals whenever i have the chance to. If the vital is placed on the south just move away and you will lose that vital and get a new one. At around level 3-4 I reset the lane. The most important thing about doing the most damage in lane is the AA reset. Make sure that you hit someone when you use your Q because it will have way lower CD than using in a spare space.
General Combo: Proc vital with Q/Auto Attack/AA reset with E/Proc another vital
Also it's very important to properly use W because in some matchups like against Jax or Riven that one skill wins you trades and it's the difference between dying and surviving. At level 6 you can kill most enemies once you get them to 50% of their HP.
After level 6 you will have your Tiamat which also serves as an AA reset, and you can proc vitals even faster. When you are against champions like Garen and Trundle it is very important to know when to pressure and go in and when to back up.

In this phase of the game you should have at least 20% CDR and even maybe 30% CDR if you have built Ionian Boots. I get this boots 75% of the time, because having 30% CDR on Fiora is a must. At this point of the game you should push side lanes and TP if the team needs you. If you don't have TP and your team plans to do an objective you can group with them. Fiora is a pretty good splitpusher because she wins almost every 1v1 at this point and if your team wants to play with splitpusher pretty easily splitpush.
This is the most important rule about team fighting with Fiora. If the enemy ADC isn't next to you DON'T CHASE THEM TO ULT THEM. This is a very common mistake that the players new to Fiora make. You focus the target closest to your team and that dives your team (usually the enemy team jungler or top laner), or the enemy with lowest HP. With your true damage you melt tanks and if you proc 3-4 vitals your team will be healed fully (more than Soraka heal) and that can be game changing. I only use ult on the enemy adcs if they are next to me, pretty much out of position and if I am sure that I will kill them, because if you chase them and they survive you can lose the teamfight.

You should splitpush at this point of the game and make TP plays. You can also make TP flanks and you can melt ADCs with 4 items and your ult, it's almost like a 1 shot. If you can't flank don't chase the enemy adc and focus the tank that dives your team as I said previously. At this point a good W makes the difference between won and lost teamfight. If you get a good W you can stun a valuable target and even if you don't dodging the stun can win you the teamfight.
If you decide to splitpush make sure that everything is warded and if you have 3 items they must send at least 2 people to shut you down because Fiora wins almost every 1v1 late game. If you splitpush you can use your Ultimate for the additional move speed and get away, or if one of the enemies is low you can turn on them and if you kill the one fast you will win the 2v1 and get a double kill.
I like to teamfight more than splitpushing because a good ult heals more than Soraka and will surely win you the fight.