As for the biting, yes to everything Richard said.

Redirect the puppy's "wrath" to toys.

If the puppy gets too wound up, pick it up by its armpits, stare it in the eyes and hold it there. Talk to it firmly and C-A-L-M-L-Y saying "no, calm down" and keep holding it like that until it stops struggling in submission and calms down.
Then you quietly set it back on the floor and go on with your routine. If the puppy gets too worked up and growly, repeat.
(Ed shows this procedure on a little baby puppy in his Dominant Dog DVD - works beautifully!)

This is the handling procedure for a young puppy. Dominance issues in an adult dog are a whole other matter, and you can read Ed's articles on how to deal with that. The Dominance DVD is also excellent for adult dogs.

Depending on the puppy's age and size, your 8-year-old might be able to do that as well if you teach him/her.

Again, you didn't give us much information on the dogs (age, breeds, temperament) so I don't know if your children can handle correcting the dogs themselves, even under your supervision.
This is just something that you will have to research first, do yourself, and then decide whether or not it's safe for your kids to try out.

At least you're looking into this while the new dog's still a puppy; it'll certainly make things much easier than waiting till it's full grown and then having to deal with issues!

First I just want to say thank you again for all your posts.

I decided to train the puppy but have him play with my kids. Once he has a command down like hopefully sit this week then I will allow them to go ahead and help more by having them have the puppy sit before he eats and have them put his food out for him. Plus take him out to potty too if it's not raining! But for now I decided not to have them do that until he has it down pat so it's not hard for my little ones.

I also have them carry him if I cannot keep my eyes on the puppy while with them the entire time. I will be in the same room cooking dinner and looking at them every 5 min or less but if I can't actually be WATCHING the puppy with them the entire time I don't have him just wander because I know my kids won't watch him like a hawk like I would like so he doesn't have accidents.

By the way the older puppy is about 15months old - Shihtzu/Poodle and the new puppy is shihtzu/maltese. They actually have the same mother and they are both males.

They play well together and we watch them when they play just in case it gets rough. The new puppy actually though is a big bully!