Basically I'm trying to figure out how much captains are supposed to worry about passenger comfort in regards to turbulence.

There are different categories of turbulence, defined by ICAO.

Light - I don't really care. It's pretty negligible. No one else cares, apart from US airlines who trip over themselves to report light chop and request a level change. It's relentless.

Moderate - Belts on, service suspended. My restraints have to go on. We have to slow down a little to turbulence penetration speed to prevent a sudden jolt causing a momentary overspeed. (High altitude + heavy = MMO/VMO is not much more than cruise speed). If there is a level change available, up or down, we'll consider it, thinking about the ramifications on fuel used, different winds meaning different groundspeed, higher altitude causing less margin to overspeed, stuff like that. My concern as an Airline Pilot in moderate is the cabin spilling hot drinks, idiot passengers standing up and ending up with serious injuries, cabin crew injuries as they secure the cabin before they get a chance to strap in, and primarily how our airspeed margins are looking.

https://en.wikipedia.org/wiki/Coffin_corner_(aerodynamics)

Severe - defined as large, abrupt changes in attitude and altitude with large variations in airspeed. There may be brief periods where effective control of the aircraft is impossible. Loose objects may move around the cabin and damage to aircraft structures may occur.

You don't really see severe very often in clear air. You'd associate it with flying though CBs which you don't really do, wake turb from a Cat D aircaft, or as a result of mountain waves, for which you'd identify on the wx chart during the pre-flight brief, receive sigmet warnings about, and might be able to visually identify with Lenticularis clouds and avoid. I don't think I've ever experienced severe. The upper boundaries of moderate definitely but I've never had pax injuries or physical damage or a significant loss of control.

I wonder if airlines have a number of planes in reserve waiting to fill in, if another plane is stuck somewhere due weather.

Yes, airlines often have 'spare' aircraft to accommodate disruption, and crews on standby (Americans call it reserve) at home/in hotels/at the airport, on call for a certain time period during which the company can call you out to operate.