dream theater is my favorite. John Petruicci and Mike Portnoy are easily the greatest guitarist/drummer combination I have seen, and although I've only been around for 18 years, somebody I know (forget who, lol) thinks they're the best combination since Hendrix/Mitchell...powerful words from my father (now i remember, lmao), who is a huge Hendrix fan. I only had to show him the symphony of Six Degrees of Inner Turbulence to get him hooked, and now, to my surprise, he's listening to things like Panic Attack and Sacrificed Sons with ease. Now I'm trying to get my 60 year old father into bands like Kalmah (lol)...I don't think it's possible
Anyways, I'm probably not as an experienced dream theater fan as many on here, but I never really liked Images and Worlds. Pull me Under is, imo, their most overrated song, I mean, I like it, but to me, Dream Theater is all about how they're the gods of musical instruments and singers...not necessarily how popular and catchy their songs are. I don't think you need to be ridiculously heavy in order to be considered "sick". Having said that, Falling into Infinity was my first, and probably my favorite DT album. Burning my Soul and You, not Me will keep you awake, especially since you said you like heavy stuff like that, while Lines In the Sand will probably give you a heart attack when you hear the cool-sounding Petruicci solo. Trial of Tears is just a great song in general, it's got it's highs and lows, but overall, just phenomenal song.
New Millenium, the album's first song, is just a great song. The beginning is great, once it starts sounding like something like Kashmir by Led Zeppelin, except better.
Other worthy mentions are basically the entire Octavarium album. The title track may be 24 minutes long, but to me it feels like 30 seconds--i wish it could go on longer. Oh, and one thing about DT is that if you're skipping through the album, it's usually worth it to listen to those really long songs that start out really slow--those ones are usually the best. Systematic Chaos is another great album--Forsaken's got great keyboard and is just plain fun. The Dark Eternal Night is quite possibly ONE of the greatest guitar riffs I've heard in a metal song. Plus, it kind of shows that they've got a sense of humor. But I won't ruin that part for you, it always makes me laugh
Also, In the Presence of Enemies pt 1 AND pt 2 are also terrific songs. Can't go wrong with Dream Theater, the big DT. Enjoy!