I think that if you want to jump start an investigation of classical music with the primary focus on finding music with the greatest immediate enjoyment factor, newbee offered the best suggestion: compilation discs with lots of works by different composers.
many booksellers with cd offerings, such as barnes & noble , borders, etc, have a selection within the classical music selection that are compilation cd's of best loved melodies, etc. borders even makes a few under their own label. these cd's offer excerpted music - sometimes entire movements or 2-5 minutes of a movement. i suggest seeking out a 2-4 cd box set since breadth is the object for you at this time. the cost will not be large. as you may know, there are also "greates' hits cd's by composer. there are also compilation discs by mood or period (ie, music to chill, baroque music for brunch, music for driving the autoban... etc)
when you have the cd's, then just start to listen. BUT do so systematically! how so? i suggest sitting down with a pen and post it. divide the post-it into 3 sections: (one post it for each cd)
a. love it (the music captivated you, floats your boat...)
b. ok or no opinion
c. dislike / not into it
when listening to each cd, jot the track number down into one of these 3 categories. dont bother with the composer or name of the work at this point. listen to the cd 3 times over the course of a week. use a different post it for each time you listen. why 3x? because your mood when listening may greatly influence your engagement with a pc of music. if you listen to everything 3x, you will give each musical excerpt a better opportunity to speak to you... or not. that is the goal here.
at the end of the week, compare your 3 evaluation sheets for each disc. you will probably have 6+ works in the "love it" category. look up the composer and specific composition. seek out these full works in your library. for more investigation, the comp discs should also have the composer's birth/death dates. maybe even the period and nationality. you may find that a large grouping of the music that has engaged you the most falls into time frame chunks which may coincide with fairly codified "periods" (baroque, classical, romantic, late romantic, impressionist, modern, etc). or nationalities, or genre (chamber music, soloist, large orchestral, etc ) . as you learn more about what engages you most, you may come to realize that you not only enjoy , for example, colorful large orchestral works, but more specifically "romantic period" large orchestral works, and even perhaps large orchestral/romantic period /russian works (ie tchaikovsky, borodin, rimsky-kor., etc).
you may find your enjoyment follows more than one distinct tendency.. .or none at all. these is over 500 years of classical music, much of it recorded quite extensively. you are not going to figure out what you like best from such a wide selection in a week. in fact, what you find engages you most this year may not be what you favor next year ! the objective is start with what engages you most readily now and continually broaden your experiences when you have the time. once you take a week or so to start your "survey" with excerpts, you will some direction.
other: wikipedia online often lists the major compositions of all the important composers within the bios, so you can get an idea where to start in terms of each composer's major works that are on your "love it" list in addition to the excerpted work. fyi: some composers are easier explore than others. for example, tchaikovsky's major works in the genres of orchestra music and orch+soloist total about 15 works, , whereas with mozart, you can easily explore 40 masterpcs in these 2 genres alone. some composers are an abundance of riches, others less so, but it hardly discounts their worth to you. afterall, you are after works that ultimately float your boat.
if you start with the compilation discs, you will not get overwhelmed, which will be easy to do if you should happen to pick up any "how to build your own music library", which is not your stated objective.
a note of caution: i would not be too concerned at this point with the ensemble or soloist performing the work. most well known and well recorded groups do a more than adequate job at bringing a classical work to life.
at this stage of introduction to classical music, whether you hear heifetz or hilary hahn or any of the other 2-3 DOZEN top tier violinists perform the tchaikovsky violin concerto (for example) is completely unimportant. most if not all have the capability to engage you fully... and float your boat.
happy sailing ! marc