It really doesn't matter. Early morning should be the least crowded for both hikes. People just don't get up early and get with it. I would probably do AL and The Narrows like at 6:00AM. Both hikes are great. The Narrows aren't strenuous at all and might feel great after hiking AL. I don't normally hike with a stick, but I would have two for The Narrows. You will want to rent shoes(maybe even a wetpants) from Zion Adventure Company for The Narrows. Allow at least an hour for this(possibly more).

Of course, do check with rangers to make sure that water level is ok and that thunderstorms aren't threatening. You DO NOT want to be in The Narrows or on top of AL with any type of weather issues.

I really like Whiptail Grill, just outside the park for dinner.
Lodge is good too.
It's nice to sit under the big tree just outside the lodge for lunch. You can see people on top of AL from there.

I'll second spirobulldog's suggestion of renting shoes from Zion Adventure Co. The rocks can get very slippery, and having the shoes really help. We rent sticks (one each) and are always very happy to have them. ZAC makes you watch a video before renting equipment for the Narrows hike, which is helpful in getting acclimated your first time. They'll also have the latest ranger updates about weather, and if it's potentially dangerous, the park will not allow you to hike the Narrows. That happened to us last visit due to massive rain and flash floods (in July/August), but luckily the next day was beautiful.

How long to spend there is really up to you, but make sure to factor in the return hike time. For me, I wouldn't do Angel's Landing and the Narrows on the same day unless I just wanted to splash and take it easy after AL. But that's partly because our lat Narrows hike was actually quite challenging in parts. The current was extremely strong that day, so we were fighting that as we explored. But if you start Angel's Landing early in the morning, I'm sure you'd be fine doing the Narrows in the afternoon. Last visit we spent close to 6 hours in the river and went deep into the canyon. But you can spend even 2-3 hours and have an incredible experience.

Was just there two weeks ago and the couple who got on the bus at the top of the loop had rented their gear.

They loved sloshing through the Narrows, with the water sometimes over their waist - and you do need the proper shoes and "wetpants", and at least one hiking stick/staff.

wetpant or wetsuit, probably depends on what time of year you are there and how acclimated you are to swimming in cold water.

Due to the thunderstorm issue mentioned by spiro I'd do both hikes in the morning. The storms typically build in the afternoon, with July and August being peak months. As spiro noted, finding yourself up the Narrows or on Angel's Landing with a storm around would not be good.

You might luck out and be there when there is no storm activity, but if you are making plans then plan to do them both early.

A side benefit of doing the Narrows in the morning is that is when you get the beautiful golden glowing light that you see in good photos. By late morning that soft light is gone, if that matters to you.

Have a great trip, Zion is awesome!