Ok. First off, as others have pointed out, the situation needs to be very clearly defined by you and the mom. They are HER kids not YOURs. This might grow to be a successful family and you become the dad, but that will happen with time. Establish how discipline with be handled, and establish it before you move it. As far as your relationship with them goes, don't go at it like you're going to be 'cool'. They will eat you alive. Don't be a dick, but establish very clear boundaries of what's expected behavior wise, and respect in regards to you. Kids will adjust, but they'll also push the boundaries too. It always easier to go in there being very strict and authoritarian in the beginning and loosing up as time goes by, than trying to transform from cool guy to 'ok lets lay down some ground rules now'. This cool guy first move ALWAYS fails. The kids will warm up and grow to respect you with the first approach, and eventually love you if you're around long enough. As others have pointed out, kids are awesome in their ability to adapt. If you're reliable and there for them, they will grow to love and respect you. It's the mothers that you have to look out for. I don't want to be the turd in the punchbowl, but you do realize that the odds are heavily stacked AGAINST this working out, right? You've got a mom that has kids with lots of daddies (don't forget there will be a time when you'll have to deal with that too. If you were a dude and you knew you had a kid out there, wouldn't you eventually seek the kid out? I know I would) which more than likely might make you as just another 'daddy' (whether you produce kids or not) to eat up and consume. I know you're in 'love' right now, but when it comes down to conflict, typically, this is a girl that has experience and knows how to execute the option of picking up and leaving. This could crush you. Not to mention you're own track record (getting married/divorced while really young). Don't get me wrong. I do wish you the best, and hope for nothing but happiness, but it would be a disservice if I didn't point this out to you as one bass brother to another.
Still, with that all said, I do have a close friend that did the same thing as you. The kids now call him daddy, he recently married his girl (last month after being together for 8 years), and now he's formally adopting them (dad is a tweaker gang banger in prison on a 25 year stint right now). He's happy, they're happy, and they've got a little girl of their own on their way. So it CAN work, it's just that odds are against it.
Good luck brother. Jump in feet first with your eyes open.