Throbbing Gristle is the first Industrial band. Monte Cassazo (ap?) coined the term Industrial and their label Industrial Records was home to some of the best groups in that early period.
That was in the late seventies, and a lot has happened since. The “Industrial” scene has diverged into a more aggressive, distorted version of trance known as EBM and popularized by Front 242 on one side, and a heavy metal with programming and distorted vocals approach known as Industrial Metal on the other side.
There are bands who continue to make true industrial music, which I define as “Experimental electronic music with a rock aesthetic and an emphasis on sound design over melody”, but they are usually categorized as Noise for some stupid reason.
Some good examples of this are Wolf Eyes, Nautical Almanac and The Sightings.
Like someone else said, Throbbing Gristle were before Industrial Metal and while they did use guitars, they did not do so in the traditional sense (as in playing chords etc.) but more as a tool for sound generation.
Ministry were among a handful of acts to really nail down the Industrial Metal sound, but it’s hard to say that they are the first.
Godflesh is arguably the true originator of the Industrial Metal sound. You’ve gotta check these guys out if you want to hear the really primitive beginnings of the genre. (Check out their first EP, ‘Streetcleaner’ to hear what I’m talking about…simply brutal.)
Another early Industrial Metal group was Big Black, Steve Albini’s band (the guy who produced Nirvana’s ‘In Utero’ among other things). I would recommend ‘The Rich Man’s Eight Track Tape’ album for a good overview of their career.
Another candidate is Mortiis, who supposedly used drum machines at ultra fast tempos with guitar even before ‘The Land of Rape and Honey’ came out. Haven’t heard much of his early stuff though, only heard a more recent CD which totally sucked ass.
Another guy to check out is Foetus aka JG Thirwell. He was one of the earlier Industrial Metal pioneers as well.
Personally, I think Laibach, an early Yugoslavian industrial group, was the first to conceive Industrial Metal. They used a lot of orchestra sampling to create bombastic industrial rock compositions, captured on some of the earlier released around '83 and '84. (Check out ‘Rekapitulacija’, which captures their material from 1980 - 84 or the live CD of the Occupied Europe tour to see what I’m talking about.)
Their influence can blatantly be heard in bands like Rammstein and Hanzel und Gretel, which are basically huge ripoffs of Laibach. I would avoid their later material though, because after the first singer killed himself, some dorky Slavic Frank Zappa-looking dude took over with this ultra cheesy voice and it just sounds like shit to me.