There are tons of websites where you can learn what kind of questions to ask. Google "nanny interview questions" and "nanny qualifications". Look at Care.com to see the types of nannies that are out there. Do a few searches here for suggestions of qualifications. Honestly, just do searches - I know this is new to you, but it's not new for someone to want to hire a nanny, so tons of threads and websites already exist with questions, quals, salary comparisons, sample contracts, etc.

Your pricing is totally reasonable.

I have a toddler and an infant and want to make the move out of a local daycare center (reasons better left for another thread) and need to find a nanny.

The basics are -
40 hrs/wk Monday-Friday, overtime opportunities too
food prep - breakfast and lunch
driving toddler to/from preschool (if I can find one) or other kind of things like library story time or playground (so, clean driving record)
Fluent in English (a bonus but not required would be bilingual in English/Spanish)

Where do I even start? Do I just post a job on these forums? Is there some sort of website I can read on how to go through the process in terms of background checks I should perform, interview questions I should be asking, and templates for a contract? Or is there a service I can use to help me find one (but not pay out the nose for)?

Also we currently pay $17/hr at our daycare for 2. Would it be reasonable to offer $17 or $18/hr to a potential hire? I can't offer much more than that because I will have to pay additional for preschool expenses (trying to do maybe a half day program just a few times a week).