I've gotten to live through this pain before and it sounds like you're getting good advice from others, so I'll just offer a little perspective from a different angle.

Whatever you do, don't get overwhelmed and don't let better be the enemy of good. Among your 60 leads are probably 20 total duds, competitors, or lookie-lous. Among the remaining 40 might be a handful of leads that your top sales person (that's the CEO in a startup...) will instantly recognize as high value. Then a bunch in the middle. Get the whole team to lay eyes on the list and bite off the 5-10 leads that you must get back to asap, lop off the ones you know are worthless, and then delegate the homework & legwork on those in the middle while you iron out the process and get the workflow/systems in place.

Before you put any system in place make sure the CEO is 100% bought in and going to invest the time to monitor and drive follow up. When I was a CEO I "lived in" two systems, one being the CRM to monitor leads/opportunities. It has to have support and engagement from the top.