As long as you don't do anything obviously bad like open up a power supply (you can be electrocuted if you touch the wrong capacitor) or wear a woolen jumper (risk of static electricity) or spill coffee on the components, there's hardly any risk of causing permanent damage. The most common and most frustrating failure mode is that you switch the system on and nothing happens.
Processors are usually sold with a heatsink and fan. (“Open box” or “OEM“ packages might not be, but they're rarely cheaper anyway, at least where I live.) The bundled cooling solution ranges from acceptable to pretty good; you only need to swap it if you're planning to overclock significantly or you want an extremely quiet computer (for a merely very quiet computer, the stock cooler is probably ok).
If the processor is sold with a cooler, the two might already be affixed together, in which case you can directly attach the whole thing to the motherboard.
Otherwise you need to apply thermal paste (also known as thermal compound or thermal grease) between the processor and the heatsink. The layer of paste should be as thin as possible. The purpose is to fill in the tiny irregularities in the surface of the processor and heatsink, because air is a lousy conductor. Since it's impossible to get a perfectly smooth surface on the processor, the aim is to have part direct processor-heatsink contact and part procesor-paste-heatsink. There are tons of sites and videos about the process on the web, of varying trustworthiness; I don't have a particular one to recommend, but you might look at a few, keeping a critical mind.
If you need to flash the BIOS to support the processor, you won't be able to do it with that processor. So unless you have a motherboard that can flash a BIOS from USB without a CPU plugged in (I think a few high-end models can do this), you'll need to first plug in a supported processor.