The classic measure of tread depth is a Lincoln head penny. Put the penny in the tread with Abes head in the tread. If you can see the top of his head it is time to change. Also you can check to see if the wear bars, built into every tire in the US are flush with the tread surface. If the tread wear bars are even with the tread, in many locales you could get a ticket for worn tires, much less be unsafe.
If your tires are wearing on both outside edges, that usually means you have underinflated tires. If the center wears, overinflated. If just one edge, then alignment.
Tires with little tread get mighty slippery in rain so be warned.
I am notorious for driving until the tires start to spark, that is when the steel tread is peeking out. This is NOT SAFE, by the way. Also notice if your tires thump any. If they thump it could mean there is a weak spot in your thin tread and you will eventually have a blow out. Been there done that, stupid.
$100 to mount and install tires seems a bit high but then I haven't ever had that problem. I think a friend of mine had it done for something like $15 per tire. If you bought yours from mail order, perhaps they have a list of shops that discount the mounting? I think Tire Rack and the like does that.
Good Luck,
Jim Henderson