Its got a lot to do with the period the composer lived.
Ie Handel and Mozart - they lived at a time when musical instrument development was such that notes could not last very long, hence lots of tinkling up and down. (Plus the style of music at that time.) Its not for me. I am not a major fan of Bach for that very reason. Running up and down the damn scale. Too simple.
I'm afraid I have to disagree with this. The vast majority of Handel's work was (late) baroque in style and as such bears little comparison to the classical work of Mozart, and to dismiss Bach as "too simple" is crazy.
But each to his/her own.
OP said:
I know I don't like lots of classical music because it falls into one of the following categories: twee, pretty, overtly religious, melodramatic, bombastic, or overly intellectual/tuneless.
I know what you mean by intellectual/tuneless and I'd therefore disagree with previous recommendations of Prokofiev because he wrote a lot of that stuff (squeaky violins, dramatic percussion and shit like that) although the violin concertos are worth a go. I tend to skip through movements until I find one I like but find him quite challenging.
My advice would be to check out some of the more mainstream stuff to get an idea of what you like. Mozart is an excellent starting point and you'll recognise a lot of what you hear which gives added interest. All of the piano concerto's are superb but the most famous and arguably finest is No.21. Also listen to his clarinet, basoon and oboe concerto's, all of which should be recognisable and
Eine Musikalischer Spass (A musical joke) is the theme to the BBC's Horse Of The Year show.
Try Bach's Toccatas and Fugues for haunting organ music and you'll recognise the very famous "D minor". Go for Handels Water Music and Music For Royal Fireworks for uplifting orchestral pieces and the Arrival Of The Queen Of Sheba for a wedding favourite. You want to avoid religious stuff but Vivaldi's Gloria and Dixit Dominus, played loud are choral pieces that'll raise the hairs on your neck. Just imagine it's a band rather than a choir.
Posts could go on forever with recommendations. The thing with classical music is that there's so much of it and a lot of it's crap. Just like now. There's no way you'll like it all or even most of it so you need to persevere, but check out the "popular" stuff like the above first and then work around the composers to find their contemporaries to sample. It can become extremely absorbing.
ETA> I've only skim read the thread but if Danny La Rouge has recommended Schoenberg, ignore him. The man mastered the art of "random foolishness".
As did Schoenberg.