What are some effective methods to smoothly introduce a new pet to existing pets at home?

The traditional way is to first introduce away from the home where none of the animals have already staked a territorial claim. Meet at a park and go for a walk with everyone together. Let them sniff each other on leash then if all seems to jive then can look for a place to play off leash such as a friend's backyard or dog park that has a smaller area. I don't like to let dogs who don't know each other have too much ability to get away from me in case something does break out. If all seems to go well then you can migrate home. I always make sure toys and chews at home are completely picked up before a new dog comes in. I don't want anyone to have anything to guard. We start in the backyard then work our way into the house. New dog is usually dragging a leash around for the first bit in case I need to correct. Never know if the newcomer is house broken or what might get its attention that it doesn't need.

I will caveat that this is the traditional way of doing things. You will have to know your own pets and how they do best. I have one dog who is leash aggressive and I personally can't do leashed meet and greets so I let her be off leash in a yard while I bring the new dog up to the other side of the fence. If they meet through the fence good then I will proceed to take the new dog inside the fence so they can get acquainted. If you are worried that one dog might be aggressive then meeting with a barrier can be useful to give you time to read their body language with each other but it also slows their brains down and forces them to consider other options.

I couldn't add anything to this excellent answer to your question. Not only did Jenn answer the question, she gave the alternative for when things don't look like they're going to go right.

Only thing I could do is give a green bar for good advice ... for excellent advice!