I suppose I should preface this by saying there is no TL;DR here!
What is the best way to go about with the touch up?
As always there is no one way to go about this.
There are basically two options when it comes to colouring wood, colouring it directly, i.e. staining, or the use of a coloured finish of some kind. Obviously a combination of these can also be used.
Regardless of the usual preference for colouring the wood directly1 the original appears to have been done with a coloured finish so coloured finish it is.
I often wouldn't recommend it but I think you'll be best served by using "gel stain" here. Don't be fooled by the (stupid) marketing name, this product is not a stain but instead a coloured varnish, artificially thickened to some level of gel consistency2.
You'll need both this and a clear topcoat, either varnish or lacquer. Lacquer would mean a spraycan since you don't have spray equipment I'm sure. Varnish, particularly polyurethane varnish, is also available in spraycans but I wouldn't advise it here.
Spraycan varnish could work well for quickly applying finish to the top of the seat backs, but it won't easily allow for minor touchups here and there throughout the set which we can see from just the one photo are going to be required. And more importantly it would be tricky to refinish the tabletop this way (in addition to working out quite expensive).
Instead of the spraycan version I would recommend you buy a tin of standard oil-based polyurethane. It must be oil-based for what you'll do next, which is to convert it (in batches as needed) to wiping varnish. Pretty much everything you need to know to get up and running on how to make, use and finish off wiping varnish is included in the pages from Bob Flexner reproduced in my previous Answer to this Question.
Although you'll be minimally covering the finish that's still present, there will be a certain amount of this inevitably since the bare areas run seamlessly into areas where finish remains. The basic rules for overcoating an existing finish are covered in a recent Answer. It boils down to a simple catchphrase, clean and matt, but read the previous Answer through before you even shop for supplies.
- Cleaning agents. Basically dish soap in warm water is enough here, but use a commercial product like Krud Kutter if you have it already and you like how it works.
- Scuffing and/or sanding supplies. My advice: don't skimp here. You won't need a lot and whatever you get will last throughout this one project and although worn won't be in any way worn out when you're done (you might actually get months and months of use from the right products).
- Gel stain. Unless you get very lucky and one colour happens to be a really great colour match you'll need to get more than one colour. Then either use them sequentially or mix a custom batch which you arrive at through testing3.
- Polyurethane varnish, gloss only. Do not be temped to buy satin or semi-gloss because the existing finish is not super shiny.
- Mineral spirits. You'll need this for general cleanup, brush cleaning (IF you use any brushes, this project won't actually require any but you can use them if you wish) and of course for converting your poly into wiping varnish.
- Copious amounts of paper towels or lint-free cloth rags. See safety note below.
- Brown paper. Brown paper from grocery store bags is good enough for this.
Brief summary of procedure:
Step 1, clean everywhere you'll need to apply finish.
Be thorough. Work in sections, don't start at one end of the set and work through to the other. Rinse residue when you're done using clean water and wipe dry. Afterwards be sure to allow plenty of time for the bare wood to dry out! A full day minimum.
Step 2, scuffing or light sanding.
This is to prep the wood for new finish, dull the existing finish adjacent to the bare areas so it can successfully accept new finish, and to feather out the edges where needed.
Step 3, apply "gel stain" using paper towels or lint-free cloth.
This is the step that will require the most artistry, since you'll both be applying colour and wiping it away (standard methodology for "gel stain") while each time trying to seamlessly blend it into the surroundings. Don't expect the job to be done in one step, even if you luck out and find a single "gel stain" that's just the right colour plan to do at minimum two rounds of colouring. It's much easier to add colour than to remove it.
Note: judge the colour wet! There may be some or a lot of dulling when "gel stain" dries, with consequent colour change but the wet appearance will be restored when the varnish goes on.
Step 4, once the colour steps are thoroughly dry it's time for wiping varnish.
On the seat backs and the tabletop I would advise a minimum of three coats and 4-5 would be better. Elsewhere on the chairs just two should be sufficient (and just one rarely looks right), but add more if your patience holds out.
Step 5, bringing the appearance together.
After the varnish has had a chance to harden up sufficiently (a week or more) you will very lightly abrade the surface to bring the surface finish in line with the rest. A vigorous scuffing with the rough side of brown paper followed by a buff with a clean cloth might be all that's necessary here, but if the gloss level is very starkly higher a gentle abrading with 0000 steel wool first is what I would advise4.
Paper or cloth wet with both gel stain and varnish are a fire hazard. They can ignite on their own without an external spark or heat source if disposed of carelessly.
Dry flat on the floor or draped over a railing. Once dry and stiff they are safe (and legal) to dispose of in your home refuse.
1 It's preferable for a number of reasons, one being that it is more durable because it's in the wood with a clear finish on top acting as a protective layer. As opposed to a top layer containing all the colour which when it wears away reveals bare uncoloured wood, as is the case here and on much modern commercial furniture (read modern as: since the early/mid 20th c).
2 Some are more goopy than others. There's no firm reason to prefer one consistency over another, other working properties are more important.
3 Measure the mixture carefully and write it down in case you need to mix another batch to complete the job, as well as possibly for future touchups. Don't expect a small batch of your mixture to remain usable in the container you store it in, unless there is no airspace or you can replace the air with an inert gas.
4 You can do this with fine Scotch-Brite but I find the action of 0000 steel wool superior personally even though it's slightly messier to work with. Note that opinions definitely vary on this and if you would prefer not to have to deal with the fine steel dust that 0000 generates by all means go with Scotch-Brite, it sure lasts longer!