If you want to change the type of symbol (and also the color and size) used for each group, follow the next steps:
First, click on any of the data points; this will select all of them. Second, click again on the same data point; this will now select solely the points of a group. Finally, double-click on the same data point to open a dialog box that will allow you to change the color, symbol and size for all points in that group.
Graph > Scatterplot… : With Groups is not the only way to get a stratified scatterplot. For instance, if you have drawn a scatterplot from Graph > Scatterplot …: Simple, you can also add a stratification variable by first double-clicking on any of the data points and then choosing the Groups tab in the dialog box that appears, as shown below:
Notice that the result is the same as the one obtained when stratification is performed directly using the commands: Graph > Scatterplot… : With Groups.