In my opinion, go with PostgreSQL.
I've used all of the mentioned databases, but they all have their shortcomings -
Firebird is okay if you need embedded db, otherwise it lacks in everything;
SQL Express is great, but if you need to run it on linux, macosx or another non-windows system, you're out of luck; also, SQL Express is limited (it's like a demo) and when you want to (or need to, because of the limitations) upgrade to a 'real' SQL Server, then that costs *big time*;
MySql .. erm, don't even get me started - you probably love your data too much to lose it someday, right?

;
MS Access - you already know of it's shortcomings;
PostgreSQL - ever since I've stumbled upon it, I've been using it for all of my needs, and it hasn't failed me yet - support is also great (professional and via mailing lists). Oh, and talking about backuping - you have several options available, so you're very likely to find one which suits your needs exactly. And if you're wondering - yes, it can be silently installed along side with your own app.
Regards,
Tadej