Your child should have soccer shoes.
A bold statement. Let me qualify that assertion.
First of all, as I stated on the Equipment for Soccer Page, soccer shoes or cleats are not a requirement of FIFA. I recommend them because I have seen some bad spills on wet grass and muddy fields.
Tennis shoes simply don't get the job done when your child is playing in less than perfect conditions.
Trust me. Your child will play on some very wet and muddy fields - well unless you are playing indoor soccer.
Of course you should refer to the local soccer league rules. They may require cleats.
Second, my approach to this is as a frugal parent. I know you can't believe it but it's true.
You can't play soccer in baseball cleats but you can play baseball in soccer cleats.
This is what I have done in the past with my son & what I recommend for U8 or younger players.
(Actually if you don't mind cutting on a shoe you can make a baseball cleat acceptable by cutting off the toe cleat.)
Below are two picures.
The picture on the left is of soccer cleats and the one on the right is of baseball shoes. You can see the missing toe cleat on the soccer cleats.
Hopefully by the U10 or U12 level your child will be over the baseball fetish & will devote his/her time completely to soccer. Right?
A good pair of shoes IS important, but not critical. After all, soccer is about using your foot, right? You should purchase a good, solid pair of shoes. Soccer shoes or cleats (called "boots" in other parts of the world) are also designed to give a person better feel and control of the ball whereas tennis shoes frequently have bulky laces and tongues.
As equipment for soccer goes, cleats can be relatively expensive. Check your local discount stores first (or use my soccer clearance page). There is no need to buy the best soccer cleats until your child can wear them for more than one year OR unless they will be getting serious use out of them on a travel team.
Check out BigToeSoccer. They have very competitive prices with Soccer.com and a huge selection of soccer boots on sale all the time (usually previous year's models).
Another great place to look for soccer cleats is the online site, Amazon!
Here are some quick results below:
Soccer-for-Parents is excited to now offer our Simple Soccer Rules to parents "on the go". This is our own free android app (sorry ipeople) that installs in seconds and contains everything on our normal rules page along with a link to the official FIFA rules and USYSAA rules. Here is the free app!
(ad free version here)