Are you trying to keep it traditional or do something a bit more fun?

i've done many of these chairs. remove the seat, pull out all of the staples underneath the cushion and remove old fabric. save it. you will use it as a template for the new fabric. cut it to the same size, and attach using a staple gun, pulling it taut and even as you go. plenty of tutorials for this. (velvet fabric looks great on these chairs)

for the wood portion, if you want a coastal thing, use a chalk paint. just give you chair a good cleaning w/tsp, rinse well, dry,,,and lightly sand w/150-220 grit sandpaper block. you just want to knock down any rough spots and any super shiny spots. wipe clean. now paint. AnnieSloan has Provence, Duck Egg, Old White for a coastal look.

go to pinterest and search, "chalk paint caned chairs" or chalk painted French Louis cane chairs.

they even tell you how to cover the cane

So glad you found this dilemma Beth. I knew you'd know exactly the right technique. I was going to suggest chalk paint as well if Robin wanted to paint. Robin, I think coastal chic would work wonderfully for this chair. I'm not sure what you want to use for fabric, but I ran across this Kravet Latika fabric on a rehabbed chaise and I thought it was just stunning. I think either could work for a coastal chic look.

those are gorge kat. either will work w/Provence,,Old white, Linen, and the top one would look great w/Florence

Thanks Beth. There's a little shop by me that has a storefront full of chairs and other furniture done in exactly the style you show. I ran across that chaise and another chair done in the seafoam and fell in love. Even though it's a traditional pattern, I think the colors keep it updated and I think it's a great transition for an older piece of furniture to move it to a more contemporary look.

To add to above--when stretching fabric, be symmetrical. Tack it in the center of the front and back and then work toward the sides, alternating.