In my opinion, the German Army was the best fighting machine of the First World War, at least in the beginning. At 1914, the German Army made it's business being the best army it could possibly be. Even though every other country did the same, Germany was arguably the most successful.
At the start of the war, (I.E. The Summer of 1914) the German Army had the best equipment--The most machine guns , lorries, hand grenades, etc etc. Even the German uniform was superior to some it's counterparts--Especially the French (whom still insisted on wearing bright red trousers and blue coats to battle.) With the obvious exception of Russia, the German Army was also the largest at the start of the war. Being able to call up about 6,000,000 million soldiers; which was more than any other individual power could muster (Again, with the obvious exception of Russia.)
Still, Germany's success in the first phases of the First World War was it's tactics, not just it's equipment. Although every country expected war, only Germany had the best concrete plan (Even the French Plan XVII was rudimental at best.) Germany's early strategy was not just the success of the Schifield Plan which brought them just before Paris, but logistics. Germany's railways, especially those built after 1870, were designed with the idea in mind that they will be used for future mobilization. Even the majority of German army garrisons were stationed on the West per the Schifield plan.
Still, The British B.E.F. was a formidable force. But I honestly favor the Germans because of their better strategy. Pulling the "German strategy card" may be a little cliche. But it's true. The poor British strategies in the Somme, Flanders, the Dardanelles and even it's blunders in Africa come to mind. Britain's real superiority came through with it's navy, who managed to keep the German navy in port for almost the entirety of war.
Unfortunately, World War One was a slaughter fest. Being the best army in such an era wasn't much of an achievement--It should be obvious why. "As good as the Germans were" or anyone else for that matter, World War One in the Western Front was a bloody stalemate until the very last months. Because of this, the top 3 or 4 armies in the time periods were able to stave each other off because of trenches, poor tactics and machine guns. So, in my opinion, it isn't really "who was the best" but "who was the worst" which becomes a better question.
But I digress. Again and in summary, my beliefs that the German Imperial Army was the best because of equipment, tactics and logistics.