The FM Pro Server is the best way to do this.
You may want to consider using a Professional FileMaker Hosting provider, such as ourselves:
We provide the server hardware, FileMaker Server Advanced licenses, upgrades, monitoring, etc. And you can deduct services 100% on your taxes, versus having to depreciate hardware and software over a number of years.
Feel free to let me know if you have any questions either here or off-list to jmay(at)pointinspace.com. Thanks!
"The database will need to be accessed from several locations on the same network - one kiosk terminal, and then one or two admin locations."
While both of those options work and work well, given only three locations all on the same network, they may be far more than you really need here.
You can install FileMaker Pro on all three machines, then Install FileMaker and your solutions on a fourth machine. Open the files on this one machine, the host, then use Open Remote on the other machines to link to them. Using FileMaker Pro as the host supports up to 9 such simultaneous users. That's not to say that using Server isn't a nicer way to go as it offers better security, performance and back up options, but given the difference in cost, you may want to consider this cheaper option--you can always "scale up" to a server at a later date if you find you need/want to.