How to choose a contractor?

[Messmariazing]

I have no idea where to start on picking someone to do it

You need to create a big list of contractors and weed them out. Start the list from:

1. Word of mouth recommendations
2. Google Reviews
3. Angie’s List
4. Yelp
5. BBB.org
6. Home Advisor

They're not all good contractors here though. Word of mouth is the way to go. Chances are your friend or your neighbor or someone of your relatives recently had a similar problem. Just ask people.

I don't even know what brand of heater I want (most of them seem the same to me!) so I can't even pick from their dealer list! Where to begin?

As someone said in this topic, Rinnai is a good brand. Rheem and Navien are great too. We chose Rinnai for our place from this great guide on how to choose the best tankless water heater.
They review propane and electric tankless water heaters too. Check them out.