From this photo we can see you have cotterless cranks, probably square tapered, which you'll be pleased to know are still a very common interface. Yours look like they're held in place with a 14mm nut easily removed with a socket wrench and crank puller.
I don't think there's any modern, cartridge bottom bracket with this style of nut, so this will be a cup and race type. You can typically remove the non-drive-side cup with a large shifter wrench and assuming the drive-side is left-hand threaded you can remove the fixed, drive-side cup using this one neat trick.
Depending on the desired chainline, if the cups/races of the bottom bracket are smooth and without pitting or gouging you can continue to use this BB with another set of square taper cranks. If the bearings are pitted replace them (they're almost always 1/4").
The four important factors in choosing a bottom bracket are:
- Shell diameter
- Spindle length
- Interface (i.e. square taper)
This crib sheet is useful.
Front chainline is determined by combination of the BB spindle length, BB offset (i.e. spacers, asymmetric spindle) and the crankset. The crankset manufacturer will typically recommend a spindle length (or see this database).
As for a replacement crankset, you can buy new (basically any road double crankset square taper is likely to fit; or there's a few manufacturer's making quality, vintage-esque cranksets, e.g. Velo Orange), however you can also find some great quality stuff at your local bicycle swapmeet/bike shed. Whatever you choose to do, I recommend getting something with replaceable chain rings, so next time you wear your chainrings down you can simply replace the chain ring and not the whole setup.
Be aware that when your chainring teeth are this worn you have likely accelerated the wear of your chain and rear sprockets. You quite likely need an entire new drive train.
On chain itself, this is dependent on the number of cogs on the rear. With 5 on the rear you're using 3/32" chain, but modern 8-speed chain fits just fine (this is often sold as 6/7/8 speed chain).