You may not have known this, but Japanese children, like any other children, like to play games together. Here are a few phrases to get your inner child a-playing in Japanese.
いっしょにあそぼ！
issho ni asobo!
Let’s play together.
あそぼ should have a う at the end, but it is often dropped in speech, making it a shorter sound.
何してあそぶ？
nani shite asobu?
What do you want to play?
Literally, “what-do-play,” this is a common way for kids to start play.
何かしようよ
nanika shiyou yo
Let’s do something.
When nothing seems to be happening, this is the question to ask the gang. The final よ is for emphasis.
あそびに行こう
asobi ni ikou
Let’s go play.
When suggesting to go to the park or pool, this is the phrase to use. The に is used to show the reason (to play) for going (ikou)
かまって！
kamatte
Pay attention to me!
From 構う kamau meaning “to mind” “to care about” or “to be concerned for”
Games
Your English romanization of かけっこ is listed as kakkeko, by it should be kakekko.
Thank you, it was already corrected.