I'm curious what I would need to do about my mortgage if I moved from my current house to a larger house in five or ten years. Would I need to refinance my residential mortgage into an investment mortgage (at a higher rate)?
Did that twice. You don't have to do it. I was told that mortgage company expects you to live there at least a year before you can convert it to an investment. No idea how that is enforced. But i am sure if you lived there for a few years and made your payments and keep making the payments you'll be fine. All they care about is that you make your payments.
You may not need to refinance, but this debt will certainly affect your ability to get another mortgage for the new house. Intending to have the old house produce income is nice, but there is no guarantee that it will indeed be profitable, of that it won't spend significant amounts of time empty between tenants, or that it won't suddenly need major repairs, all of which could potentially affect your ability to pay a second mortgage.
This is certainly not impossible; many people have done it... but it may not be as easy as you expect. Get some quotes and run the numbers.
I should also note that the whole idea of a "starter house" is rather bogus. It's much more efficient to start by buying the house you really want, if you can afford it.