Combat in Africa was much, much more mobile, and generally amounted to large-scale skirmishing and raiding more akin to the latter years of the Boer War than the trench warfare in Europe. This was the result of the German commander in German East Africa's (Paul Emil von Lettow-Vorbeck) decision to fight things out to the best of his abilities to tie down as many Entente resources as possible.
Like Germany's colonies elsewhere in Africa, East Africa was a massive, sparsely defended region surrounded by Entente colonies and very easily cut off from the outside world. Because of the vast distances involved and the fairly small numbers of troops available - particularly to Lettow-Vorbeck - combat in East Africa very quickly turned into a war of maneuver. Lettow-Vorbeck started the war with a small colonial garrison (Schutztruppen) of local Africans led by European officers and equipped with largely outdated equipment. The Schutztruppen were equipped with obsolete Gewehr 1871 single-shot black-powder cartridge rifles at the outbreak of the war, and they had little artillery and next to no real logistics like you'd expect even from pre-railroad 19th-century warfare. They'd salvage some guns off of the scuttled cruiser Konigsberg that had gotten trapped in the Rufiji delta, and, in the absence of pack animals, Lettow-Vorbeck's forces largely relied on porters who were usually pressed into service. The colonial forces they went up against - British colonial troops (primarily Indians), South Africans, Belgians, and Portuguese - generally had similarly poor equipment at the outbreak of war, but as the campaign escalated their numbers and equipment vastly improved.
After a few initial engagements, Lettow-Vorbeck's forces were very rapidly outnumbered, and they ultimately resorted to a kind of fighting retreat. Lettow-Vorbeck's tactics generally amounted to pillaging their way across the countryside and outmarching the enemy while resorting to the occasional ambush of pursuing forces. The idea was to use scorched-earth tactics to massively strain the logistics of the numerically superior enemy pursuing them while moving quickly to avoid encirclement and setting up ambushes to keep up pressure and keep the Entente from neglecting the theater. The Schutztruppe gradually fought their way south through some pretty horrific conditions and eventually made their way on into Mozambique. There, they would continue to raid and pillage, getting their hands on fresh equipment (most importantly clothes and modern smokeless-powder rifles). Eventually they made their way back north along the eastern coast of Lake Nyasa/Malawi and ended the war just inside the borders of Northern Rhodesia (Zambia), surrendering two weeks after the Armistice.
The combat itself was generally very low intensity - fighting was limited to brief skirmishes and ambushes, and the few incidents that could be called pitched battles were very early in the campaign. For the most part, it was a grueling marching campaign through the jungle for all involved. Something on the order of a quarter million Entente soliders and half a million porters were tied down by a force that was never more than 25,000 men and down to fewer than 1,500 by the time they surrendered, and the majority of the casualties in the campaign were from disease and starvation rather than actual combat. Often overlooked is also the toll on the locals - Lettow-Vorbeck's scorched earth tactics were immensely successful, but they were also devastating to communities it passed through, and both sides relied heavily on the often-forced labor of porters. While military casualties were only around 60,000 for the campaign, over a quarter million civilians died thanks to famines created by the war.