The major problem one faces when having multiple pets is keeping them from having scuffles, normally these fights are playful and the pets mean no harm but many times owners need to keep their pets away due to the numerous fights. The problem is most evident among cats where one or more would reject another. It is very difficult to live with these kinds of internal conflict but it is easy to make sure that there is no bloodshed.

Animals like us live according to hierarchy/ranking and cats & dogs have different rules of hierarchy.

1. Dogs being descendants of wolves always live in packs (groups) which is led by one Alpha (male or female). Typically the oldest or largest being the ring leader so scuffles for leadership are always present whether playfully or aggressively. The best solution? make sure to keep your pets pampered according to their age where the oldest gets their food first, their leash put on first or given a bath first. When you reinforce this sort of control, the younger brats realize their place in the pack.But when setting the ranking among your pets, you need to remember that you are the Alpha female/male so if you have 2 dogs the ranking should be you, older mutt and then the younger mutt. This prevents the older mutt from being aggressive in front of the younger one.
2. Cats are primarily solitary creatures who never live in groups but can live with one or two partners at the most (siblings or mates most commonly) but upon introduction of a new cat they can get hostile and sometimes run away from home. To prevent cats from fighting, you can either use their age to give them their ranks or keep the hostile cat in a separate room away from the others (A special spot where only it is allowed to sit/sleep).
3. Have a mix of cats and dogs? The only method is to use their age and never break out of ranks. My dog is the oldest while one of my cats is the youngest so they are given treats, fed & pampered according to their age and they seem to understand that. The best time to get them used to the ranking system is when they are kids so if in case you introduce more pets they are not too bothered by it as they would know their ranks.

When they have their scuffles (dog vs dog, cat vs cat or dog vs cat) do not interfere unless they get serious. And when you do interfere do not scold or beat anyone but simply separate the pets until they have calmed down. Sometimes there are pets that cannot get along with the other no matter what, the only solution is to keep the most aggressive one away from the others (this includes the water and food bowls).