Banjo strings are thin, cheap, and change the instrument more than their price suggests. There are only a few decisions to make and one of them is specific to this instrument.
What gauge changes
Tension, mostly. Lighter strings bend easily, respond to a soft touch, and sound thinner; heavier strings drive the head harder, feel stiffer, and give more volume for the same attack. Bluegrass players tend heavier because the instrument has to compete; clawhammer players often go lighter because the attack is a strike rather than a pull.
Gauge does not change the pitch you tune to, and it does not change any fingering — every shape in the library is identical whatever is on the instrument. It changes how hard you have to work and how the head answers.
The fifth string is its own decision
It is a different string doing a different job: shorter, tuned higher, and almost never fretted. Sets pair it sensibly by default, but it is the string that goes dull first because it rings on every single beat regardless of what the left hand is doing, and it is the cheapest single thing you can replace when the banjo starts sounding tired.
When to change them
When they stop responding, not on a calendar. A banjo tells you sooner than a guitar does, because the head amplifies exactly the frequencies a dead string stops producing. Change the fifth string on its own between full sets — it is one string and it does the most work.
Material, briefly
Almost every banjo string sold is steel, wound in nickel or bronze on the lower two. Nickel reads a touch smoother and lasts slightly longer; bronze reads brighter and dies faster. Nylon or gut is a genuinely different instrument — lower tension, far quieter, and the sound of a nineteenth-century banjo rather than a modern one — and it needs an instrument set up for it, so it is a decision rather than a swap.
The one thing not to do is mix a heavy set with a slack head. Tension has to be balanced across the whole instrument, and a heavy set on an under-tensioned head just presses the bridge into the skin and kills the response — which is a setup problem being blamed on strings.
After a change
Stretch them, tune, play for ten minutes, tune again. New strings will not hold for the first while, and on a banjo the drone drifting is far more audible than any other string drifting. The tuner is free and in the browser.