iggypump said:
I want to make this experience as easy and affordable as possible.
It might seem like the easiest, cheapest, and fastest option would be to get married as soon as possible and then apply. But in fact this option will end up being difficult, expensive, and slow.
It is better to meet her a few times, spending as much time as possible together, before getting married. At the very least, do not get married the first time you meet in person.
Your application will be processed much more quickly, and you are more likely to get an approval, if you first take the time to get to know each other in person. Appeals can take two years, and cost thousands of dollars. Do things now such that you have the highest chance of being accepted.
One tip: the visa officers expect the stepparent to know a lot about their stepchildren, and to have as good a relationship with them as is possible under the circumstances. Make sure your girlfriend knows a lot about your children, and that they get to know each other - even if it is just conversations over Skype or whatever.