In Acrobat Pro 6, the form tools are available from the Tools > Advanced Editing > Forms submenu, or you might want to turn on the Advanced Editing and/or Forms toolbars. Consider turning on and snapping to the document grid to make laying out form fields easier (you can configure the grid to match up with the blanks on the sheet if it measures out evenly; I got a fairly decent match on your lines with 11 gridlines per inch and a vertical offset of .16"). Don't neglect the Properties; if you want a predictable "look" in the filled-in fields, you need to make sure they're all set to a uniform text size, at least, instead of the default "Auto" (experiment with Shift-click and click-drag to select multiple form fields, to save time by setting lots of properties at once).
It's true: typed form data cannot be saved in Acrobat Reader, unless the creator has Acrobat Pro 7 and makes it so the form *can* be saved in Reader (this *might* require Reader 7, but I'm not sure of it).
If your players are willing to invest in add-on programs for Reader, here are a couple programs to consider:
CutePDF Form Filler ($30):
http://www.acrosoftware.com/Products/CutePDF/Filler.asp
PDFill Form Filler ($10):
http://www.pdfill.com/
Having never used either, I can't personally vouch for them, but if they do what they claim to do you might have a solution here.
Depending on your needs, you might simply print the completed forms to another PDF (losing the form fields, but keeping the data). For anyone that doesn't have Acrobat Pro, this means installing a PDF Printer driver; make sure to print comments (look around for it in the Print... dialog):
pdf995 (free):
http://www.pdf995.com/
primoPDF (free):
http://www.primopdf.com/
Make sure you don't enable any security features before doing this; it might interfere with printing a PDF to a PDF. I've never used these drivers, so I'm not even sure if it'll work properly (and it's certainly inferior to retaining the form fields), but that's another way to go.