To get a cheerful family, you'll need to get a "cheerful" adults by only using the second item for training. It doesn't matter which one you get- some examples are Memetchi, Uhyotchi and Mumutchi (there are seven cheerful adults in total counting Makiko).
After obtaining any of the cheerful adults, you'll need to mate another cheerful adult. Just make sure that you don't do what you would normally do to get a pure Meme family or you will get a pure Meme family and not a Cheerful family.
For example you could mate Memetchi and Uhyotchi, or Mumutchi and Watatchi. It doesn't matter which two cheerful adults, just as long as you get Mememamatchi and Memepapatchi as parents.
Hopefully they have one or two eggs. As far as I know there is no surefire way of getting a certain amount of eggs (with the exception of things such as pure families and the space characters). There is about 30-40% of getting one or two eggs. If you do get one or two eggs they should both hatch into Pantsutchi/Omututchi and you'll have a Cheerful family. Unfortunately, if you happen to end up with three eggs (which is more likely) then you're stuck with another blended family.
I hope that made sense and didn't confuse you too much!