The first car-telephone
From 1910 on it appears that Lars Magnus Ericsson and his wife Hilda regularly worked the first car telephone. Yes, this was the man who founded Ericsson in 1876. Although he retired to farming in 1901, and seemed set in his ways, his wife Hilda wanted to tour the countryside in that fairly new contraption, the horseless carriage. Lars was reluctant to go but soon realized he could take a telephone along. As Meurling and Jeans relate,
"In today's terminology, the system was an early 'telepoint' application: you could make telephone calls from the car. Access was not by radio, of course -- instead there were two long sticks, like fishing rods, handled by Hilda. She would hook them over a pair of telephone wires, seeking a pair that were free . . . When they were found, Lars Magnus would crank the dynamo handle of the telephone, which produced a signal to an operator in the nearest exchange." [Meurling and Jeans]
Thus we have the founder of Ericsson (external link), that Power of The Permafrost, bouncing along the back roads of Sweden, making calls along the way. Now, telephone companies themselves had portable telephones before this, especially to test their lines, and armed forces would often tap into existing lines while their divisions were on the move, but I still think this is the first regularly occurring, authorized, civilian use of a mobile telephone. More on mobile working below.
Around the middle teens the triode tube was developed, allowing far greater signal strength to be developed both for wireline and wireless telephony. No longer passive like a crystal set, a triode was powered by an external source, which provided much better reception and volume. Later, with Armstrong's regenerative circuit, tubes were developed that could either transmit or receive signals. They were the answer to developing high frequency oscillating waves; tubes were stable and powerful enough to carry the human voice and sensitive enough to detect those signals in the radio spectrum.
More on ho w a triode works and its history is here
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Resources:
[Meurling and Jeans] Meurling, John and Richard Jeans. The Mobile Phone Book: The Invention of The Mobile Phone Industry Communications Week International, London, on behalf of Ericsson Radio Systems (1994) p. 43. ISBN Number 0952403102
More on mobile working: Johan Hauknes points out that "L.M. Ericsson had already developed telephones for military purposes in the field -- mobile -- I would guess of the same kind as Meurling and Jeans describes, tapping into fixed systems. That's according to according to Ericsson's Centennial History which is written in Swedish."
"LME [sold] a large number of transportable field telephones and so called cavalry telephones to South Africa during the Boer War from 1899 to 1902. Several types of transportable telephones for military purposes had been developed by LME during the 1890s, bought by the Swedish Military. This according to Messrs A. Attman, J. Kuuse, and U. Olsson, in LM Ericsson 100 år Band 1 Pionjärtid - Kamp om koncessioner - Kris - 1876-1932 (vol. 1 of 3), published. by LM Ericsson in 1976."
"Finally, the first transportable phone documented in the centennial volume is from 1889 - primarily for 'railroad and canal works, military purposes etc.' There's a facsimile of an ad of this in vol. 3: C. Jakobaeus, LM Ericsson 100 år Band III Teleteknisk skapandet 1876-1976.) Railroad related maintenance and repair work, such as for signbased telegraph systems, was a major source of income for LME in the first years."