Which lacrosse shoes are best, really depends more on the field you will be playing on than anything. Most games are still played on grass but with the explosive growth of synthetic materials and artificial surfaces even down to the playground level the choice of shoes can be very important.
The newest long blade synthetic turfs act and play very similar to natural grass fields without the slippage you get from slightly muddy or wet grass. For either surface a good ¾ inch cleat is a good safe choice. Although there are some excellent choices available now, you do not have to purchase lax specific shoes if you allready have a soccer , football, or basketball shoe that will work.
Standard soccer or football plastic molded cleats or the slightly more expensive replaceable cleats work great for grass and long turf fields, but they cannot be metal cleats. Typical football cleats tend to give greater traction on the ball portion of your foot for maximum grip, and I prefer these for defensive players who are going to be making a lot of contact and need that extra traction while they are pushing. Soccer cleats are generally a little lighter and have shallower and often fewer cleats generally positioned in straight lines to allow good cornering ability while still allowing some slippage while kicking through the grass to a soccer ball. Because they are lighter I prefer them for attackmen and speed type kids. This is really just a personal preference and by no means am I a shoe expert, but this opinion is simply based on my own observations. Any plastic cleated athletic shoe will work fine especially at the lower levels. The most important thing is that the shoe fits well. I also personally prefer the high top or Mid top shoes, because they provide greater support of the ankles. They are a bit heavier, and restrictive though, so many kids choose the low top versions for speed and agility.
For box lacrosse or for the older generation artificial surfaces, the shoe really can vary. I have seen many players wear plastic cleats on these surfaces and they work just fine. In other cases the kids wearing cleats tend to slip and regular tennis or basketball shoes work great. There are also specially made turf shoes that have many tiny cleats covering the entire bottom of the shoe. These also seem to work very well on all of the turf surfaces.
My best advice is to ask someone who plays on the fields where you will be playing and see what the other kids are wearing. If possible watch a game or two and see if the kids are slipping or if their traction seems to be good. If they are slipping on a turf field and they are wearing cleats go with regular basketball or turf shoes. If they are slipping on natural grass and the field is excessively wet or muddy then you may want to look at going with a cleat that uses longer than ¾ inch cleats. Or purchase the ones that are replaceable. And most importantly make sure they fit well.