Apocalyptica got me started in my favorite genre (and related genres) of music for the past two years. I simply love Apocalyptica.
Enverxis's and zhentil's suggestions are quite good. As for my suggestions in addition to what they've mentioned...
I wouldn't disregard
E.S. Posthumus's Unearthed. It's on the classical side of things with deep and moving melodies. The album is also good in that it varies in tone from track to track (some are dark, some are softer and more Roman-styled). I often like to equate the sound style with
Gladiator's (the movie) soundtrack.
... now onto music that aren't classical but is very similar to Apocalyptica.
Apocalyptica's genre is either Instrumental Rock or Symphonic Metal (if you look through shops and music sites). I think Apocalyptica is more related to the second genre however since I personally haven't found many artists in Instrumental Rock that has Apocalyptica's characteristic emotionally dark and deep sound. Most Symphonic Metal is a different from Apocalyptica since it's a subgenre of Metal however. That means a "metal percussion-ish" sound with lots of lyrics. Note however there are a number of Symphonic Metal artists that have a strikingly "symphonic" melody performed in a way to emulate a large classical group. Artists that employ female vocalists usually have an almost "opera-ish" characteristics in regards to the lyrics. As for the specific artists, I believe
Within Temptation and
Nightwish resemble Apocalyptica the most in the Symphonic Metal genre. I for one was first pulled into the genre by Apocalyptica and subsequently now love
Within Temptation and
Nighwish as well.
In truth, I can't say if you will like those two bands or not though I bet most people who like Apocalyptica will like them. I for one used to listen to solely classical music until Apocalyptica introduced me to Symphonic Metal. As a matter of fact, I don't like normal metal and rock at all. Hahaha.

While Apocalyptica is classical music that has metal influences, Syphonic Metal is metal with classical music influences. The reason I recommended those two is because those two bands are two of the most classical-ish artists in Symphonic Metal.