You need an amp that has a broad frequency range. Guitar and bass amps are not sufficient. Keyboard amps are a bit better and there are a few dedicated drum amplifiers from Roland, Simmons, and Alesis. They might be adequate for practice but in my opinion will fall well short of what you hear through headphones. Your best bet is to run the DTX module through a really good PA system with a sub-woofer, but I realize this may not be cost effective.
Total beginner here, I would like to connect some speakers to my Roland TD-1 set, i tried plugin-in an old guitar amp and it sounds HORRIBLE - especially the bass and the Hi-Hat (The sound of the stick hitting the rubber is much stronger then the sound the amp is letting out). Are there any low...
Home: headphones (or a PA if you *really* need speakers)
Rehearsal: a PA
Live: a PA
Minimum of 12" woofers, speakers on stands, tweeters at ear height. Test by playing Prodigy's Firestarter thru at full pelt.
(as for make and model - that depends on your budget)
Avoid cheapo things with Bluetooth and don't be swayed by things advertised as "drum" amps as most are just overpriced active floor wedges/speakers with a badge stuck on.
Oh, and if you see something that says 2000W output - see what its power consumption is (clue: it won't be 2KW... The output is actually 350W RMS max)