Assume 100 years from now we have established sustainable bases on Mars, to the point where a colony ship making the trip to Mars is not "breaking news" anymore. There are a few reasons to begin to colonize locations such as Ceres (ignore the reasons not to), but what other locations in the asteroid belt are worth our time and energy?
Also, would there be any resources in the asteroid belt worth guarding and restricting others from?
The big reason to go to the asteroid belt is to get valuable materials. I believe that the most valuable materials to be found are volitile compounds such as water, nitrogen-rich ammonia, carbon dioxide, or methane. These things are not strictly abundant on Mars, and you may want more of them. Especially if you are making a biosphere via terraforming, you will likely want a lot of ammonia to process into nitrogen for an atmosphere, carbon dioxide to process into oxygen, and water.
The amount of metals or other minerals that industry would demand would be insignificant compared to the amount of materials needed for terraforming. While some metals are very valuable, if you did mine a lot of those metals, prices would drop and might make them less profitable. Demand for rare metals just isn't that high right now compared to the amount available on Earth, the Moon, or Mars. Regarding volitiles, if space based colonies around Earth or elsewhere in the inner solar system are a thing, they may have demand for ammonia (for fertilizer), water, and carbon dioxide as an oxygen pre-cursor. This could produce high demand, even if there is not the very large demand from terraforming Mars.
Given that you want those materials, it would be best to mine them and process them into frozen balls, then shoot them towards a Mars orbit where they would be captured and processed as desired. The closest place to Mars to get such materials would be the asteroid belt. Farther away locations, like the moon systems of the gas giants, might have better sources of materials (i.e. more concentrated, especially for methane), but transport costs will be higher due to distance and the gravity wells of those planets.
Thus, you will want industrial facilities in the asteroid belt to process materials for shipment to Mars. Here is where automation matters. Some would argue that these facilities could be completely autonomous. I think that is pretty optimistic. I believe that there will have to be stations of maintenance personnel to manage these industrial stations where asteroids are broken up, separated to release valuable elements, re-packaged, and then transported off. There is a lot going on with these processes, and equipment breaks. You don't want to have to launch from Mars (potentially years away) to fix problems in the belt.
Furthermore, I don't think that completely autonomous vehicles doing the mining/crushing/packing is realistic either. I think they will be partially automated with human 'operators' controlling them; the way drones are operated today, or like the construction equipment is pictured in Avatar. If that is the case, you certainly won't be able to control the vehicles remotely from Mars. The lag time for signals would be several minutes. In that case, you will need even more personnel living in the asteroid belt as remote asteroid crusher drone operators.
Once you have a working population of thousands, it becomes efficient to have some 'cities' or stations that serve as commerce and entertainment hubs for the workers in the belt. Again, the distances involved in taking leave back on Mars, much less Earth, are prohibitive.
There is no reason to concentrate these people on Ceres. Ceres' surface gravity is 3% of Earth's, not enough to provide any medical benefits for humans living there. Humans would be much better off living in larg space habitats that can rotate to create their own gravity. Some planets or large moons might not have enough gravity to support humans, but would still be valuable for agriculture. Mars is most likely to fit this criteria. Ceres' low gravity would probably preclude even genetically engineered plants from operating properly (regarding internal water transport, specfically) even if kept in pressurized greenhouses. Ceres' low insolation doesn't help. Even farming for these residents of the belt is best done in space stations.
There isn't a lot of motivation to live in the belt if you are not either working there, or operating a business or service for those working there. There is a lot better real estate in the solar system for things like space-based farms and solar energy. If you were going to have space colonies, then you would want them much closer to the Sun, perhaps even closer than Earth, to take advantage of food production and cheap energy.
Thus, your overall population would be low, just enough to support the industry. This would be much like mill towns of the Industrial revolution, like Lowell, MA or Blackburn in the UK.
The materials most likely to be in high demand are volitiles like water, carbon dioxide, ammonia, and methane. If this demand is high enough, the asteroid belt is one of the best places to meet this demand for the inner solar system. Unless you believe that the industry for refining and shipping these materials will be fully automated, you could expect the asteroid belt to be littered with 'mill towns,' clusters of space stations and space farms that support the operators and maintenance techs of the asteroid belt industry.
Given that the belt has materials both in abundance and in easy to access form (both in terms of deltaV to get to and from asteroids, and in terms of the asteroids themselves being mostly small bodies with limited gravity), the real answer is people will preferentially go to the belt.
One of the key costs in space transportation is energy, so having to fight large gravity wells (like that of the planet Mars) imposes a cost. Materials on Mars are there, but generally you will be travelling long distances to get to places where water ice is available, or particular mineral ores or even just places which are useful or aesthetically pleasing to live in. You will also have just enough gravity to be annoying and just enough atmosphere to get in the way, but not enough (like on Earth) to sustain an ecosphere without a lot of heavy duty terraforming.
The asteroids are divided into several types, with the "S" class asteroids generally being made of silicates and other "stony" materials, "C" class dark carbonaceous objects and "M"-class metallic asteroids. (there are lots of sub groupings, but to be simple we will stay with the three main classes). This provides materials which are generally useful for all space endeavours, including building and shielding materials, metals and volatile elements needed for life support. Large quantities of water are also thought to be contained on most types of asteroids, either as ice, hydrates or bound to other compounds.
Mars is perhaps a "sentimental" choice, and I'm sure lots of people will initially go to Mars because it is interesting and exciting, but as the solar system becomes more settled, market forces will impel people to go to where costs are lower, materials are easier to access and conditions in habitats can be custom built, rather than contending with alien atmospheres and gravities (an asteroid's materials can be repurposed into building a proper space habitat). By the end of the 22nd century, I suspect Mars will be a sort of quaint backwater while the smart money is heading out to asteroids and the plethora of small moons (essentially captured asteroids) in the outer reaches of the Jovian system.
Even solar energy will be easier to access in the Asteroid belt, since all that is needed is an arbitrarily large mirror (or platoon of mirrors) flying in formation with the asteroid, while solar energy on Mars needs to contend with the day/night cycle, effects of the atmosphere and gravity or jockeying platoons of mirrors in orbit around Mars to get the energy desired where you want it.
Go the the asteroids, young man.
If this world has a bit of fiction mixed in you could easily add a new element that could be very valuable or possibly have a new particle that could conduct its own energy making it an effective energy source if processed. Thus there would be motivation to establish mining facilities, which would be inhabited by workers, that operate machinery, that collects the resource. The resource is then shipped to desired location via cargo transport. The transports open the door for pirate raids but I did like your unique idea of shooting resource "balls" to be collected in orbit of desired planet. I hope this helps!
I don't think the asteroid belt itself is really a practical source for resources unless you are desperate. The asteroid belt for all intents an purposes fairly empty as the number of astroids(in the trillions(~10^12 probably) is drastically less than the total net volume even assuming that these astroids only esist within the area of a disk 1 AU(about 149597871 km or (1.496E8) from 2.06 to 3.2 AU.
The area gives pi*((3.2)^2 - (2.06)^2 )=~18.8AU^2 which is clearly much larger than a trillion(i.e. 10^12). Now recognizing that the asteroids are actually scattered above and below the plane and unevenly distributed the real astroid belt is actually even less dense! In fact the amount of mass in the astroid belt is less than 5% of the moons mass (~4% according to a brief google search).Even more critically the mass is not evenly distributed among the astroids but is actually focused almost entirely into Ceres and Vesta with the rest being distributed among the remaining 10^12 asteroids. That is almost negligible. In fact most asteroids are just tenuous bits of dust and debris that have lightly clumped together over since the asteroid belt was formed.
Think if you were on a spacecraft traveling though it it would be very unlikely to pass close by a single asteroid. You would almost certainly have to intentionally visit an asteroid meaning it isn't a task you can
For mining to be worth it economically the cost of obtaining the resources in question would need to exceed the cost of actually visiting the asteroid. This is almost certainly a big no.
The only possible exceptions to this would be Ceres and Vesta which could be useful if the economic value for their resources were to become profitable particularity Vesta due to having once been a small differentiated planetoid in hydrostatic equilibrium before it suffered a major collision.
Thus in conclusion resource mining would be highly economically unfeasible in the asteroid belt. Unless there was an extreme deficiency of a particularly common element I don't think it ever could be viable ever(except possibly tourism)
