AdamH, as an easier start can't you just uncheck 'Respect Familiar Cards' on the game setup options if you're playing against a friend who just signed up? Not ideal for now but may get you up and running more quickly.
While I like the concept of the Familiar Cards list, I think that it currently has some "discoverability" issues and needs to be made clearer as to why the early games for new accounts are so heavily skewed towards base (EDIT: A tiny selection of base, apparently). It's been ages since I set up my account, is it really only 10 cads marked as familiar? That would explain some things, I thought it was the whole base set. I think it should be. 10 is too little.
Here is what I always imagined the process for a new account would ideally be when they first sign up:
When they first log in, a window says "Welcome to Dominion Online, is this your first time playing Dominion, either digitally or the physical board game?" with yes or no buttons. Clcking Yes takes them straight to a tutorial on how to play Dominion.
Clicking No, brings up another message:
"Great! Which expansions are you familiar with?" followed by a checklist with all of the expansions named, along with an extra "I don't know" button. Maybe base game will come pre-selected.
All players will then start with all base game cards marked as familiar, but those who selected more expansions will automatically have the cards from those expansion listed as familiar too.
Finally, the first time a player clicks on the Familiar Cards tab, there is a quick message explaining what the page actually is.