I am considering a tankless water heater but am concerned about the high mineral content of the water we have.
Currently I have an old electric water heater that I flush every year and there is a lot of rust and the elements are coated in rust.
I would like to know how the tankless water heater will handle the rust, whether it will get plugged up quickly or shorten the life expectancy.

Won't the water (between the heater and your faucet) just sit in the pipes as it does now? I think the rust comes from sitting in the tank, something that will not happen with the tankless heater.

I've used tankless water heaters with hard water in several locations without any trouble at all with mineral buildup.
Rinnai and Rheem are the brands I would buy again, and you'll want to go one size up from the recomended capacity for your home.
Don't forget to ask about tax credits for installing EnergyStar equipment, it's up to 300 bucks, and you'll want to let your homeowner's insurance carrier know, as well... some offer a discount for going tankless because it reduces your risk for water related claims.