Out of state investing in cities within a reasonable driving distance may be a good option. 

Are there any markets you would recommend for long distance investing?

I think they contain solid information and will help lay a good foundation as a beginner (Haha, I'm completely new to real estate as well, btw)I'm also going to be reading the following books, and hopefully I can finish all by the end of this month:1) The ABCs of Property Management2) The Advanced Guide to Real Estate Investing3) Long-Distance Real Estate Investing How to Buy, Rehab, and Manage Out-of-State Rental Properties (this might be the most important one to me personally)4) The Book on Investing in Real Estate with No (and Low) Money DownIn addition to reading, I'm consuming crazy amounts of youtube videos, and I'recommend you do this as well.

We like the short-term rental idea because we could stay at the property ourselves when we want - summer break, spring/winter breaks, etc.Our budget will be right around $1M or less ideally, and we would like a short distance to the ocean.How likely is it that we could find a property as described above and have it cover or come close to covering the mortgage? 

Bigger Pockets is a great place to learn and connect with fellow investors.If you’re looking to invest long distance, I definitely recommend building your core-4 team.

The red flags of managing from a distance and hiring a property manager are legitimate concerns that make it more challenging.

I recommend you read "Long-Distance Real Estate Investing" by David Greene.

They should ONLY include properties of the same size (sq ftg), and within a short distance form this property.Lastly, your profit comes after you recover your cash you put in. 

If the only constraint is you don't want to manage it from a distance, why not just hire a property manager? 

With regard to the proximity to the beach, I tend to have a beach distance of 15 min or 5 mile rule, whichever is less.