How can I finish a mission faster?
Your mission is split into two separate parts: The driving and the tasks you have to do upon arrival.
When driving try not to lose any momentum by accidentally bumping into NPVs or other objects, since acceleration takes quite some time. This can easily happen when driving in curves. If you are too fast you will lose control of your vehicle and hit something. Instead of losing all your speed at once it is advantageous to brake and slow down before curves to assure smooth and fast driving.
For the actual mission, it is beneficial to know your surroundings well. Try to tie multiple tasks together. Maybe you can cut down the electricity on your way to a victim. When opening a locked door, open every other door you know you will have to go through later on too so that you do not have to retrieve the pry tool, which would cost you valuable seconds – but be careful: fire will spread through open doors!
Lastly use the AI or play in multiplayer. The workload you have to handle is more manageable with help. Let someone establish supply and attack lines for you, while you are taking care of something else. The AI can transport the victims very efficiently to the paramedics taking the shortest safe route.
It may be tricky at first (especially for more difficult missions), but eventually, with a little bit of practice, you will be able to complete your missions fast.