The early goth scene was publicised primarily by radio, the music press, fanzines and word of mouth.
Radio exposure was primarily due to John Peel, whose late-night Radio 1 show was incredibly important for devotees of alternative music, and most early goth bands got their first national radio exposure on his show. In time, some degree of "goth airplay" filtered down to other shows on Radio 1, but John Peel had invariably played them first.
The three main music papers at the time were Sounds, NME and Melody Maker, all of whom featured up-and-coming goth bands from an early stage. Two articles in early 1983 described the new bands as forming something akin to a "scene", one being the "Positive Punk" article in the NME and the other being an article by Mick Mercer in Melody Maker in March 1983 (reproduced here). Mick Mercer took on the editorship of Zig Zag magazine from October 1983 and steered that in a more goth-friendly direction, becoming something of a champion of the "goth cause" (he still is, having written several books about the scene).
The power of the main music papers (along with John Peel) is hard to underestimate. For a long period they regarded themselves as taste-makers, and also taste-breakers. Having been slow on the uptake about Punk, the music press were desperate to spot the Next Big Thing in plenty of time, and sometimes quite happy to help things along by declaring they'd discovered a scene that maybe wasn't quite a scene yet. Whether they were jumping the gun with "Positive Punk" is debatable, but I'm inclined to regard the goth scene as fairly well-formed by that point.
The downside with the music press was that their quest for novelty and the Next Big Thing meant that once they'd discovered the Next Big Thing, the Previous Big Thing was suddenly last year's news and fair game for ridicule. Goth, unfortunately, supplied them with plenty of ammunition.
Not all music journalists acted this way, of course - some of them, having felt passionately about a form of music, were not prepared to suddenly turn upon it once it was deemed unfashionable. Unfortunately, in the case of Goth, there were few jounalists prepared to stick by it - Mick Mercer is one noble exception here.
Some journalists were equally consistent in their loathing of goth, or at least certain goth bands. Bauhaus had a particularly edgy relationship with the press, at one stage (October 1982) having an acrimonious "onstage press conference" at a gig with Steve Sutherland from Melody Maker. Sex Gang Children had similar problems, with Andi Sex Gang once throwing a pint of water over a journalist.
On the other hand, some bands were very good at working with the press, particularly The Sisters of Mercy under the eminently quotable Mr Eldritch.
Whilst the main music papers revelled in being cutting edge (and sometimes just cutting), certain other publications tended to feature bands only when they were already popular - Bauhaus had most certainly "made it" once Pete Murphy appeared on the front cover of Smash Hits, though in 1982 the magazine was not nearly as pop-centric as it was later to become.
Inbetween the music papers and the likes of Smash Hits were a number of smaller-circulation magazines, including Noise! and Zig Zag. As I've said above, the latter magazine was to become noted for its goth-friendly coverage under Mick Mercer's editorship.
Goths and goth bands also occasionally appeared in style magazines like the Face (who featured the Batcave in a feature in 1984) and I-D.
At the grassroots level there were also a number of fanzines, all of which were self-financed and appeared sporadically. They were of variable quality, but the best of them, such as Grim Humour, married good writing to a refreshing lack of music industry cant. In an age before email, websites and mp3s, fanzines provided a valuable adjunct to the mainstream press.
TV didn't really come into it till 1982, when Bauhaus appeared on Top of The Pops. Following that, other goth bands appeared on TV and the BBC screened a documentary about the Batcave. Goth bands later appeared on the Old Grey Whistle Test (a BBC2 music program) and The Tube (an innovative "youth" music program from the recently arrived Channel 4).
It's hard to rate the relative importance of TV. Goth bands featured on it relatively rarely, but when they did they reached a far wider audience than they would normally get. Michael Johnson, late of Nemesis and now of Starvox, dates the period of goth's popularity from Bauhaus's appearance on Top of the Pops with Ziggy Stardust, with a later wave of converts from Siouxsie's appearance with Dear Prudence.
One last thing is worth mentioning here, with regard to publicity rather than media - the goth live scene.
The early goth bands were primarily British and toured the UK a lot, so they got a lot of local exposure. It helped that a lot of them were very good live, often much better than they were on record (the popularity of Sex Gang Children in particular is very difficult to understand without having seen them live). Indeed, some people could become obsessed, following them round the country (as happened with Play Dead). And of course people would be selling fanzines and exchanging opinions about bands at the gigs, so alongside the conventional media of radio, music papers and TV there was a strong "underground" network of live bands, fanzines and word of mouth. It is this network that created the "early scene" that was later picked up by the music press, for better or worse.