There are already some great suggestions above, although some of them are walking a fine line in fair housing violations. An owner should not involve themselves in tenant selection in any way. I also think it's important to hire a PM that is also an investor. You'd be shocked how many PMs are renters themselves and have no idea about investment strategy and that has a big impact on how you manage properties. If you are a new investor, the answers to many of the questions above probably won't do you any good either, you need to have context.
I am going to give you some unsolicited mindset advice instead- my apologies in advance!
I own a fairly large PM company and I see many of our "green" clients making the same mistake over and over. They want the PM to take on all of the responsibility of their unit, but as the owner, they want full control of what happens there- it doesn't work that way. Your PM is your business partner and, to a certain extent, you need to trust them, in many cases, for your own good. Often times, new investors create problems and liability for themselves by mucking up the processes.
You are hiring a PM because they already have a team, systems and processes in place. You should trust them to screen tenants and maintain your property using those systems, that's what you are paying for. It's counter productive for an owner to insist on reviewing tenant applications, asking for 14 bids for basic maintenance items or wanting to have access to the property or their tenants.
I'm not talking about blind trust, but understand that PM is generally a very low margin business and the efficiency is where the profit is for both of you. A really good PM will only allow you to interfere with their processes for so long before they cut you loose- but a crappy PM might let you meddle because they are desperate for the business. It's hard to know which you have if you don't have full comprehension of the business. You have to trust your PM within reason and if there are trust issues, either you have the wrong PM or the wrong mindset, but one of those issues will cause you to lose sleep.
Happy to expand if you'd like, best of luck to you in your search!