Ideally, you should have a first aid kit in your car and in your home. Of course, this doesn't mean you have to have a special store-bought first aid kit.
Putting some adhesive bandages, pain killers and Benedryl in your medicine cabinet is probably good enough for much of your first aid needs at home. Tap water is fine for cleaning cuts and clean towels can be used in a pinch for controlling bleeding.
Your car, on the other hand, doesn't have a linen closet where you can grab a clean towel or a sink to rinse your cuts. Where I live, commutes to work of more than an hour each way are common. When you spend that much time in the car, it makes sense to be prepared. This is where a regular first aid kit comes in handy.
Emergency Supplies on the Road
Besides a first aid kit, keep a bottle of water, an emergency flashlight, and a blanket in your car on long trips in case you get stranded. Make sure you have enough water for every person in the car. A gallon of bottled water is more than enough for 4 people to spend the night in a vehicle.
A mobile phone may be the most important safety item you take with you on the road. Keep it charged and pay attention to areas where service is limited or unavailable. If the farthest you travel in your car is to the supermarket down the block, then a mobile phone is probably all the safety equipment you really need.