I did have a kit in the car ages ago but it seems to have vanished, it is just a standard kit you can buy. I treated myself to one from lidls. It has everything in a plastic pockets sort of thing that lifts out, so you can see everything and get to it quickly.
I would say gloves are a must for the car one but not so much for at home. If you are trained you might want a mask for resus. The kits have it all such as a variety of dressings on bandages, swabs, triangles etc. BTW cotton wool is not normaly included or recommended, possibly due to it being to fluffy and making more of a mess of any wound.
In the house I have a medicine box, a first aid box and a box with emrgency stuff like big swabs, wash and bandage. Its supposed to make it easier to crab TBH I would just grab the whole lot.
One thing I did from early on was have a small kit in my bag for going out with the kids. Easy to treat kids cuts and grazes etc when out shopping rather than upsetting the kids trying to find a loo or get a shops first aid kit when all you need is a plaster.
I got one of the plastic boxes the plasters come in and filled it with kids plasters with pics on, antiseptic wipes, mini tube of magic cream (antiseptic), plasters and a couple of calpol sachets. Takes up very little room and has been used a lot on dys out and at the shops.