Hi Dianne! Congrats on the new family addition!
Puppy barking is a totally common problem with new additions. There are many reasons a dog will bark continuously, including understimulation, anxiety, aggression, alerting, and attention seeking. I believe most puppy barking is put down to understimulation and attention seeking. To try and control the barking, pick a one word “no” command to give whenever the puppy barks (no, quiet, enough, for example). Always give the command when the puppy is barking. Always reward the puppy when he stops barking or isn’t barking in situations he normally would be. If he is barking at you, do exactly what you said you’ve tried, and ignore him. Remember to not yell at him, he might interpret this as you “barking” back, and don’t use physical punishment. Positive reinforcement (rewarding the behaviour you want) is much more effective. Use treats and pats and happy tones when the pup is quiet. And try to provide as much environmental enrichment as possible – toys, play dates, etc, to keep him entertained!
As far as your puppy concerns, try not to worry too much! Your puppy is exhibiting nomal puppy behaviour, they are often a handful when you bring them home. They are figuring out what they can get away with and where they play into the hierarchy of your home. You must establish yourself as the dominant one from the start, by training and reinforcement, so that your pup knows to listen to you! You should start training pups from the time you bring them home, and from the sounds of it you are on the right track! Your efforts so far are great, it will just take repetition and patience I think. For training your pup, try breaking training sessions down into a lot of short sessions, like 5-10minutes. Always follow the same routines, repetition is how puppies learn what to do! And remember reward based training works wonders! Find a treat he loves and use it to your advantage!
The basics of training for anything are, as I said above, positive reinforcement, as well as counterconditioning and desensitization. This means you should expose your pup to as many social situations and interactions as possible while he’s young, so he does not react badly to them as an adult. This includes new people, animals, noises, stimuli, etc.
Just stick with your barking pup, he will outgrow the rambunctious phase and will settle quickly, as long as you stick with it. And remember, when he’s about 6 months you should consider getting him castrated, as this helps tone down unwanted behaviour as well.
Hope this helps, I’m sure you and your little rascal will get on fine