Is there an easy, built in way to bookmark a character's progress - so that if they quit then come back they can pick up where they left off?

Only option that I am aware of is to issue a password that they can enter when they start the game. Problem is, you'd have to track inventory and flags and such.

Or, they can just type save...

Is save going to work if they're playing on the public website? There's no login requirements for the games there.

you can download quest games and play them offline, there you can save. And you can save your game progress, when you play online here on textadventures.co.uk as a regiestered and logged-in user. Other website would have to install the Quest Webplayer and create their own user administration.