My advice in order to do it right:
1.) Learn about music - attend concerts (preferably unamplified), pick up classical/jazz/rock/etc. for dummies books, maybe even take music appreciation class
2.) Discover what aspects of sound reproduction pushes your buttons (deep/full bass, pinpoint imaging, super detail, crisp highs, etc.)
3.) Learn a bit about room acoustics, recommend reading Floyd O'Toole's "Sound Reproduction"
4.) Decide what room you'll be listening in - avoid square/small rooms (unless you'll be using headphones), shared rooms lead to all kinds of compromises
5.) As far as practical acoustically isolate the room (lined/flexible ductwork, insulated exterior door, insulated staggered stud walls, float an insulated ceiling
6.) Set an overall system budget
7.) Determine which approach works best for you (no right answer, see below)
8.) Shop for speakers/headphones first, power/integrated amps simply serve the speakers/headphones
9.) When trying to do serious auditioning limit yourself to 3 or 4 setups per day (aural memory can be fleeting), bring a wide selection of the music you like that focuses on those audio attributes you love, and take notes to force yourself to really listen and have a record of what you've heard
10.) Final auditioning must be done in your room
Some of the various approaches to home audio:
1.) Tubes (versus solid state) - more hobbyist involvement, tube rolling (swapping out one for another) can be expensive/maddening/tail chasing, usually lower wattage (pair to higher efficiency/more expensive speakers), care must be given to pair to synergistic speakers (bass related impedance issues), often associated with simple/pure circuitry and low parts count but several circuit approaches
2.) Solid state (versus tubes) - less involvement, more stable, easier to match to a given pair of speakers, less background noise, more efficient (less heat created), also several circuit approaches
3.) Vinyl (turntables) - again more hobbyist involvement (setting up, changing/flipping records, cleaning records), albums are more expensive/harder to find, considered by many to still be the gold standard but surface noise and vibrational feedback are it's banes
4.) Digital - nowadays computer based, many options from using your computer to play CD's you've ripped to streaming off the web (Tidal is a good option at $20/month) using ChromeCast or AirPort Express to buying a dedicated music server, various software can be added to enhance sound quality or integrate ripped with streamed music, several formats exist (and have come and gone), recommend being somewhat computer savy
5.) Various speaker types exist (although there is no perfect one) - traditional box speakers with drivers pointing "forward" (sealed, ported, etc.), dipole (open drivers mounted to a board, or a variety of ribbon materials), front or rear horn loaded drivers, omni-directional designs utilizing a sealed/ported/etc. enclosure on the bottom. This is the only transducer in a digital system (something that changes one form of energy, electricity, to another, mechanical motion) and has to do it in a relatively unknown environment to the designer, so it has by far the toughest job to do and the highest distortion levels, so it makes for the most individual preference based decision of the entire audio system.
