How did this proto-geek cuture arise?
Well, first off, it didn't come out of whole cloth. Enthusiasm for science fiction and fantasy had been building for decades, and informal associations of fans had grown up around dime novels and certain serial magazine characters like Sherlock Holmes by the 1890s. The transition to pulp magazines would come in the early 20th century; Sam Moskowitz in "The Origins of Science Fiction Fandom: A Reconstruction" wrote:
The obviously existed - as Gernsback's comments show - a considerable pool of science fiction enthusiasts, who carefully followed those magazines that made a policy of publishing such fiction, who collected the material, who were very literate and prolific letter writers to the magazines, and who visited book shows and one another. [...] [Robert Hobart Davis] made it a point to have reader's departments in all of his magazines. This policy began in The Argosy for February 1911. The number of letters enthusiastically praising the "different" stories, as Davis called science fiction and fantasy, was impressive and included the names of readers later to build reputations as writers, specifically H. P. Lovecraft, Otis Adelbert Kline, and Edmond Hamilton.
Magazines like Weird Tales (1923) attracted considerable reader-response, and a close affinity between readers and writers, with many fans (like H. P. Lovecraft, Robert E. Howard, Robert Bloch, Fritz Leiber, etc.) making the leap from having letters published in the magazine's reader page ("The Eyrie") to having poems and stories published in the magazine itself. Lovecraft and his friends were also deeply involved in the amateur press, and an argument can be made that the amateur periodical The Recluse published by his friend W. Paul Cook in 1927 was the first fan magazine, publishing considerable material from pulpsters H. P. Lovecraft, Donald Wandrei, Frank Belknap Long, Clark Ashton Smith, and H. Warner Munn - all published in Weird Tales.
Yet it was Gernsback's Amazing Stories (1926) which proved the catalyst for fandom:
It was editor Hugo Gernsback who first called the letter writers to his magazine "fans," and his remarks bear quoting again as hey appeared in his editorial "The Lure of Scientifiction" in the third issue of Amazing Stories (June 1926): "There is not a day, now, that passes, but we get from a dozen to fifty suggestion as to stories of which frankly, we have no record, although we have a list of some 600 or 700 scientifiction stories. Some of thee fans are constantly visiting the book store with the express purpose of buying new or old scientifiction tale, and they even go to the trouble of advertising for some volumes that have long ago gone out of print."
Gernsback's publication of full names and addresses in the letter columns encouraged fans to write to each other, but the formalization of fan groups was created in 1928, when readers formed the Science Correspondence Club (correspondence clubs were at the time common; amateur journalism and groups like the Lone Scouts were dependent on correspondence for geographically isolated members).
These correspondence clubs in turn led to local meeting clubs, and efforts shortly began for the different correspondence clubs to combine into a national club - and these became dues-paying organizations which issued their own small sheets or journals, the real beginning of the fan press. The first of these was The Comet (1930), put out by the Chicago branch of the Science Correspondence Club; by the third issue it had changed its name to Cosmology.
Amateur groups and publications proliferated faster than the science fiction pulps; there were a variety of cheap printing methods available, such as hektograph and mimeograph, and early fanzines were usually very primitive affairs - although some of them were high-quality letter-press works with stenciled or engraved illustrations. The fanzines are inherently fragile and many didn't survive, but you can get an idea by checking out the Hevelin archive.
By 1931, certain fans like Forrest J. Ackerman were already making themselves known on a national level for their fan-activities, especially forming and running fan clubs and publishing fanzines, reviews, fiction, etc. and getting letters published in magazines or publishing advertisements in pulps. The first fan "convention" was Philcon, held in 1936 in the basement of a fan's home in Philadelphia, which was repeated annually until World War II; although the first "big" convention was the First National Science Fiction Convention in Newark, New Jersey (1938), followed the next year by the first World Science Fiction Convention in New York in 1939.
A lot of the raw material for "geek culture" arose between 1927-1939 - including conlangs ("constructed languages") and slang, pen names, cosplay, fanart and fanfiction, etc.
Were there other fan groups which focused on radio or comic strip content?
Yes, although the nature and distribution of early radio programs and comic books was a little different. The comic strip Little Orphan Annie debuted in 1924, for example, and while you could write letters to the editor in newspapers, praise for comic strips rarely made it into print; the comic strip made the jump into a radio serial in 1930, but was only broadcast locally in Chicago, not reaching a national audience until it switched networks in 1931. Like Gernsback, the producers of Little Orphan Annie encouraged fan participation, especially redeeming Ovaltine box tops for decoder rings (remember that scene from A Christmas Story?) so that readers could join the "Little Orphan Annie Secret Society" - these were very explicitly groups created for fans rather than by fans, but radio serials for The Shadow also had a large cross-market with the pulps and developed their own fandom.
Comic books took a little while to gestate, with the earliest magazines being collections of comic strips, and some pulp magazines even running strips in their pages; the start of the Golden Age of comic books was the publication of Action Comics #1 in 1938, which introduced Superman. These early comics were very closely connected with the pulps - often with the same publishers, editors, artists, and writers - and many of the same pulp tricks appeared in comic books sooner or later, including letters pages which included the full names and addresses of fans, but it took a while for this idea to "take hold" - the first recorded comic book letter column appeared in Target Comics #6 (1940), and they weren't a regular feature of comic books until the late 1950s. Comic book fandom consequently took a while longer to organize.