Ok, presuming that you are not going to spend thousands to build a sonically isolated studio in your basement then, like most home recorders you are going to be facing compromises between perfection and budget. A typical home recorder faces several sonic challenges, some of which are enhanced by being in the basement, and some are reduced.
One problem is outbound sound transmission. In other words, your music bothering the neighbors. Sound transmission is stopped by mass. Fortunately this is one area, except for the ceiling and windows, that basements shine. If you have a problem with sound getting out of the windows and completing sealing them or upgrading to double/triple panes is not an option then get really heavy curtains for them. If your playing is driving everyone else in the house crazy then you are going to have to think about both the ceiling and the starway door. To repeat, mass stops transmission.
The next is inbound sound transmission...cars driving by, the party next door, etc. Here again basements have a significant advantage but you can still get transmission through windows so the same steps like heavy curtains will help. You can also have inbound transmission from someone playing a TV or Radio loudly above, but hopefully since this is your own house you can just turn that sound down.
The next is structural sound transmission. In other words you are right in a quiet part and you hear clump...clump...clump from somebody walking across the floor above. This is different that other transmission because the sound is actually being created by the floor. Think of the floor as a drum head that is being beaten by the feet of the person walking across it above. Like other transmission sources it is stopped by mass, but you have to pay attention to how the mass is attached so that it is isolated. For example a typical ceiling might have gypsum board screwed into the joists. Sure that is mass, but because of the rigid contact between the joist and the gypbd the sound is just going to travel through the gypdb too.
The next is mechanical noise. For a typical upstairs studio this is the heat/air conditioner radiator or fan noise. Basements are usually the location of the main HVAC unit so the noise of the whole house starts there. Basements are also the main plumbing entrance and exit points so you might get a lot of water running through pipes noise. Nice quiet passage...with a featured solo of thirty seconds of waste gurgling down the pipe because someone flushed the toilet upstairs.
Finally is the question of excessive reverberation. It is the one most often thought of when people think of improving their space, it is the one most often associated with basements because of all that concrete, and it is the one most easily alleviated. Get some OwensCorning 703 panels or rock wool rigid panels, wrap them in burlap, and hang on the walls. Repeat until your recordings do not sound like they are in an echo chamber. But remember, and this is why I mention them last, is that they do not contain much mass. So while they do stop reverberations they do not do much for transmission, and transmission is your enemy too.
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As for the question about the drum kit, mics, etc. There are two issues, one is protecting your hearing from being in an enclosed space with a drum set and/or loud amp, the other is cross-bleed between the instruments. Anyone dealing in close proximity with a drum set should consider hearing protection, including the drummer. Other than that there are 'gobos' which are freestanding panels made from plywood (for mass) and insulation (to cut reflections). Search around and you will see instructions about making or buying them.
