I could write a book about what to do and what not to do. If you have zero experience hanging a door, let alone exterior doors, then I would recommend that you find an experienced carpenter to advise, help, or do it for you. I know the focus of this website is largely do-it-yourself, but I have found that the most important thing for homeowners is correctly judging when to get outside help vs. when to tackle it yourself. You are spending a lot of money on a new door unit (or just new doors?) so you can't afford to mess it up. If you think you have water problems now, wait until you install this door or flashing incorrectly, then you could have an insurance claim.
I am occasionally tempted by the lure of outward swinging doors. I have custom built many and installed them. There are many things to consider pro and mostly con. I usually just punt on this question and install sliders. They have their own problems, but I know I can live with those. Maybe others could contribute their thoughts on this subject. I will throw out some potential problems with outswing ext. doors just to get the ball rolling; 1. The doors are more likely to be exposed to the weather and swell or warp - this is a minor problem easily prevented with adequate paint or poly on ALL edges; 2. Where is your screen door going to go - custom made to swing in?; 3. What happens when the wind catches the door? - Most of the time the door will swing too far and pull off, or bend the hinges, or damage the side of the house, or damage the door or jamb, or break the door glass - great care must be taken to install robust swing-arresting gear such as a chain with dampener etc. and/or a doorstop/hook catch to hold the door open; 4. Outswing doors demand more space outside - do you have enough room for people AND the door to swing at them? - Usually the answer for FRONT doors is no, people will be swept off the landing and not be invited in by the psychologically inviting and welcoming inswinging door.
I know all of this sounds kind of negative, I am partly playing devil's advocate and introducing some things you should consider. If the door frame and ext. flashing and threshold are in decent shape then you might want to just have the doors replaced and not the whole frame. Also, if there is a water problem then you might have some unseen rot in the surrounding framing that might have to be repaired. I can't see it from here but one option might be to have NEW inswing doors installed. If it is done right then it should fix your water problem. There is no reason why inswing doors should bring a lot of water with them - Maybe install a small, water deflecting entry roof above the doors.