LittleBikie

I have an old (folks say, Vintage) Sears Tote Cycle that I would like to update. I have several things that I want to change, but I don't know where to start once I get past the re-painting.
My current ride:
Steel frame
'Folding' bike : not really... it has a joint in the center which separates the bike into two pieces
3 speed : 1972 Sturmey Archer S3C hub with coaster brake (shift indicator is missing, so it is stuck in the hardest gear . I want to upgrade this to something newer, but not sure what to replace with. Also the gear shifter is located on the seat neck to allow for the bike to be separated into two parts.
28 spoke (7 9/16") 20 x 1.75 wheels. Again, I would like to upgrade, but not sure how much larger I could go or if I can swap out the 28 spoke with more common 36 spoke wheels.
Chain: 106 links (52.5"). I want to get a different colored chain... thinking gold or a bronze color
Seat post: 24.8 diameter. I would like to get one that is a little longer for some added height.

I will need new front end brakes and pads/shoes. The current front caliper is attached to a spring carriage that is a pretty cool thing to have.

I have been to a few bike shops, but they keep trying to sell me a new bike. I already have another bike (mountain), but this one is for Slow Rolling and short distance rides. Plus I ADORE it and it rides well.

Looking for suggestions. Thanks!

veganbikes

I wouldn't really upgrade anything on it. The most I would do is get it working. You can get Sturmey parts that are now made by SunRace that work with the older systems so if the hub is functional just get a new shifter and replace any cables and common wear items like chain (which is probably a 3/32) and regrease/repack everything. If you are looking for brake pads that will work the Kool-Stop Continentals will likely be the ticket and are the best of what you can get.

The seatpost is an uncommon size and I feel like you will run into problems with other parts. The reason a lot of shops are telling you to get a new bike is the bike as the physical object is not worth much and was never a high end bike to begin with. To you it may have sentimental value, hence why I said get it working but trying to do more is just not the ideal way to go.

Moe Zhoost

Definitely an interesting bike and well worth keeping. I, like @veganbikes, would not do much except fixing/replacing things that really need it. Sturmey 3 speed hubs are relatively easy to rebuild and parts are easily found. You won't really find much improvement in a more modern 3 speed hub. Are the rims steel or aluminum. If steel (especially with plated spokes) I would be tempted to replace them if really rusty. I doubt that you will be able to fit bigger wheels, though. For 20" wheels, 28 spokes should be adequate, unless you will be putting a lot of weight on the bike. I suspect that their weight limit exceeds that of the whole bike.

Good luck and post pictures of your progress

thumpism 

Net pic for reference. Keep in mind that if you want wheels with more spokes you'll need to find a different Sturmey hub. Sturmeys generally work fine. I prefer them without the CB but that might not be advisable for your bike.