If you are staying at a Disney Hotel, then you have Magic Hours from 8am and can ride some rides before the park fully opens at 10. If not, you can still get in to the shops on Main Street and be ready for the rope drop into the rest of the park at 10.
You have reservations at Chez Remy at 3:30, which is in Walt Disney Studios. I am suggesting that you won't want to rush from one park to the other all day because you will wear yourselves out - BUT there is more to do in the main park, and that is where the fireworks are, so if you keep that reservation you will end up going to the main park, then the studios, then back to the main park. Yes, they are close to each other but don't underestimate just how far you'll be walking. And July could be HOT. MAYBE rethink the restaurant?
Leaving that aside, work out what rides you really want to do, plan in some slower paced activities (riverboat ride, or a show in the Studios, for example) and try not to double back on yourselves. Grab a programme of shows and a map as you arrive.
If Pirates is open by July 13th, head there first, grabbing a Fast Pass to Peter Pan as you go by. Do the rides in FantasyLand before it gets really busy and work your way back to the front of the park. Remember to say hello to the dragon under the castle. The shops in FantasyLand are worth looking in, just for the theming of the buildings.
If it is still morning, get a ride on one of the horse drawn vehicles down Main Street; or pop in some of the shops. Head over to Studios for some shows - tram ride, stunt show around your reservation. The lessons on how to draw Disney characters are fun, and a nice change of pace - in the Animation Studios.
The parade is actually worth seeing, but view it near the Royal Castle Stage and as soon as it passes, head to DiscoveryLand for Buzz because there will be minimal queue then. There is also very little queue for the FantasyLand rides when the Princess Shows are on, on the Royal Castle Stage. Don't queue for a character meet, but do take time to watch some of the meetings, because they are very sweet interactions.
If you don't want to do jerky rides, then you can consider cutting out a lot of AdventureLand and FrontierLand, to cut down on walking. If you can, grab a map to look at in advance. Take food in with you to keep you going, and fill up water bottles at the many MANY water fountains.
And remember to have fun, relax, look around and enjoy the atmosphere. You'll not manage to get every ride and show done in a day, so accept that and just go with it!