Inside Paris (the municipality, which is basically the same thing as zone 1 on transport maps) and in the innermost suburbs, you can go everywhere with the metro (and in places RER), but buses provide a useful complement to get closer to some destinations or if the trip by metro would require a detour and changes. There's no real system. Most bus lines cross the city, but not in a straight line. For long distances (say going halfway across Paris), buses are typically much slower than taking the metro even if the metro requires a connection; unless you want to see the sights, you'd usually not stay on a bus for the most part of its route.
Lines number 1–14 are the metro, 20–99 are Paris inner city buses, and ≥100 are suburban buses. Some suburbs also have buses operated by different agencies with their own numbering system. Lines T1, T2, etc. are tramway lines, using the same fare as buses.
Within Paris, there's no particular numbering system. There used to be a system where the first digit indicated one endpoint, from 2x terminating at Saint-Lazare to 9x terminating at Montparnasse. But that wasn't ever much help: if you're at Montparnasse and want to go to Saint-Lazare, do you take the 28? You can, but the 95 is quicker. You can still see traces of this system, but the latest evolutions haven't kept the first-digit rule. For example, the April 2019 reform created a line 25 in the south-east very far from Saint-Lazare, shifted 24 so that it no longer runs between Saint-Lazare and the south-east but instead between the center and quite a bit further south-east than before, and many similar changes.
There's a hub-and-spokes system in some suburbs, where you take a train or metro to a stop and continue by bus. The lines tend to follow commuter patterns; I can't think of a place that has a gridlike system with north/south and east/west lines. You shouldn't expect the kind of organization that's typically found in Germany in Switzerland at least, and I think in the Netherlands too: if the train is scheduled to arrive at :00, the bus is scheduled at :01, and the train is two minutes late, don't count on the bus waiting.
All the metro stations have a network map, but bus stops don't these days. All bus stops1,2 and all the buses1,3, have at least a line map like this one. The line is all flattened, so this gives a poor idea of local geography, but you may recognize some familiar names here and there. You should be able to get a paper copy of the Paris bus map with major streets in manned metro stations. Maps are of course available on the web, in the RATP app and in many other apps.
Apps can also show you the expected arrival time of the next bus (except on some suburban lines). Most stops also show this information. Inside the bus, the name of the next stop is normally shown on a display near the center of the bus and announced vocally, but these systems aren't always working and are sometimes out of synch, so pay attention to where you are or ask the driver if you aren't sure.
Directions like “northbound/eastbound/…” are never used in Paris, all signs list endpoints (and occasionally major intermediate stops, e.g. most suburban RER stations have clear indications of which platform to go to for Paris). So you will need to check a map if you only have a vague direction in mind rather than a precise destination.
1 At least inside Paris, I'm not sure if this applies to all suburbs.
2 Temporary stops due to construction may not have a map.
3 Except when they don't, for example when a bus is reassigned to another line in a hurry.