that looks like an old cottered crank and chainring as one piece. that crank also looks really short. really, really short. cottered cranks aren't hard to find, though I've never looked for anything that short. most cranks are in the 165 to 175 mm range. if I had to guess, that one looks closer to 100 mm.
if there's
enough clearance, I would think about getting longer cranks and a bigger chainring. if you're happy with the cotters, there's no need to change that. if you're not certain about what size chainring you want, it might make sense to get cranks with a spider instead of an integrated chainring. that way you can try out a new size without buying a new crank. make sure you hammer or press the cotter in instead of just tightening it with the nut.
if you want to switch to a different sort of crank (square taper, for example), you'll have to change the bottom bracket and spindle, too, which isn't terribly difficult, but does involve either some specialized tools or the patience of Job.
since you don't have any deraillers to worry about, an 1/8-inch single-speed chain should be fine. they are cheaper and more durable than derailler chains. you'll probably want a chain tool for that job, though it does look like you've got an adjustable tensioner that might be able to take up any slack if you don't want to cut your chain.