this is just my experience and yall feel free to correct me if im wrong im a tad rusty and its late. You would be using Bridge mode if you were trying to connect to the internet thru a router. example: you have a Linksys router and a Verizon dsl modem. So, you would bridge the linksys and the dsl modem together.
Now I am assuming you just have 1 dsl modem so your ISP is correct in saying you should stick with PPPoe mode ( which does mean you are always on) but you also may have it configured to where you have to enter a username/password to connect to the internet. Some people do and some dont.
As for your ISP's claim that your modem has an internal firewall I SERIOUSLY doubt that... Most firewalls are programs such as Zone Alarm. They do have external firewalls but that is another piece of equiptment that I doubt you currently have.

In conclusion: keep your modem in PPPoe mode and get yourself some good antivirus/firewall software

try Zone Alarm Firewall and Adaware for antivirus you could go with something like AVG. hope this helps.