You have choices on how to proceed. It all depends on what you wish to do with your furniture. Do you wish to keep its character and not change anything about it, but keep it protected at the same time? Do you want it to give it a fresh new look? Do you want to colour it? Here are some options for you to consider:
The first option is to leave the finish as is and give the cane furniture a thorough wash with sugar soap and re-varnish. This will keep the character of the item with its faded parts but it will now have a new protective coating on it.
The second option is to give it a detailed sanding. You will remove all the old varnish on it and reapply your choice of stain and varnish. I recommend using the Ozito 200W Detail Sander.
Please look carefully at the chair and make sure that it is still structurally firm should you decide to sand it down. Cane bindings will sometimes become brittle as time moves on. It may be something to consider as it becomes another item that you may have to repair.
I paint a lot of old furniture and have dealt with cane furniture often. I would usually give it a light sand using 240 grit sandpaper. Personally I would do this by hand. When varnish gets old it will be quite brittle, so it should come away quite easily.
If you intend re-varnishing, best to sand all the varnish off. It'll be tedious, but worth it in the end. If you'd rather paint it, then all you'll need to do with the sandpaper is get it to a smooth finish, and won't have to thoroughly remove it. Wipe it down afterwards with methylated spirits, which will remove any excess dust, dirt and grease.
I generally just use a spray paint, such as Rust-oleum Primer + Paint, which has very good adhesion. It comes in a wide range of colours, although classic white (in Rust-oleum it's Blossom White) will team with any decor and fabric. Go for either a satin or full gloss finish so that keeping it clean will be easier.
When choosing colours to use, think about the overall decor in the room. Is there any main colour in that room? Try and pick up that colour in the fabrics, which will give the room a continuity, and make it all look intentional. - LizzieCro