If I were to send a mining machine in, for example, a Falcon Heavy, I would first have to inspect the asteroid before I could start mining, so I would really have to send 2 rockets. First to inspect and find the best landing place and then I would launch another to deploy the mining machine. Or will this all fit in one rocket (Falcon Heavy can hold 16800 kg)? Maybe this will differ from what machine I'm using but I really wouldn't know what mining machines are the best to use. As no one has gone to an asteroid to mine it, this is all hypothetical.
Chances are it'll be a lot more than two. You'll need some exploratory missions to investigate multiple potential asteroids. The first mining mission is likely to be a small-scale testbed. Eventually mining any significant amount of material off a large asteroid is probably going to require many Falcon Heavy-sized payloads to assemble a big enough infrastructure.
Your approach is probably right and it would fall neatly into the time honored tradition of using "prospectors". Typically in conventional mining (on Earth) a prospector is a single person or team of people who go out and look for resources--not to mine them but to find them. Prospectors are typically scientists who take samples, make test digs, and use other tools to locate ore or other valuable deposits. Instead of mining the deposits themselves, the prospector then essentially sells the location of the strike to a large company which has the motive and the muscles to employ hundreds and fully extract the resource.
Although mostly hypothetical at this point, asteroid mining will probably follow a similar principle. At first, "prospector" satellites will be sent out which are extremely light and have good sensor capabilities. After these satellites visit multiple targets, they analyze them, and select the most optimal targets for a later mission (for example by tagging the asteroid with a radio beacon). This is important because it's possible that an asteroid has no resources that would be worth the cost to extract and refine them. Then, mining companies could bid on the rights to that asteroid or people could send a ship to mine it.
Currently one of the only companies actively working on asteroid mining is Planetary Resources. They have plans for such a "prospector" satellite which will visit multiple asteroids and scan them to determine targets for a later mission.