From what I can gather, you're essentially asking a two-part question here:

1. What did WWI trench systems look like and how elaborate were they?
2. What did the fighting in these trench systems look like?

I'll answer this primarily from a defensive point of view, addressing both questions together as well as I can. Before I begin, I'd like to note that what follows is an overview of the topic. I'll necessarily be skipping a lot of details and painting a picture with fairly broad strokes. My answer's going to be long, so I'll break it into two pieces. Part 1 is below; Part 2 will be a comment on Part 1.

Lastly, "Axis" refers to the alliance of Germany, Italy, and Japan in the Second World War. The alliance led by Germany in the First World War is known as the Central Powers.

Okay, here we go.

Part 1: Forward Defense

Trench systems changed over the course of the war and varied depending on what army we're talking about, but contrary to popular belief, they were rather advanced and recognizable right from the beginning. For example, prewar British defensive tactics emphasized the importance of constructing a trench network comprising two lines: a line of firing trenches at the front and a line of cover trenches to the rear. Fire trenches served to provide basic shelter from enemy artillery and small arms fire while concealing defending infantry. Cover trenches, in contrast, served to primarily shield men from artillery. As a result, cover trenches would be deeper, have undercut recesses for men to hide in, and, when possible, contain forms of overhead cover. Barbed wire would shield the firing trenches. When necessary and when time permitted, redoubts (essentially a position enclosed on four sides with trenches and shielded with barbed wire) would be constructed behind the cover trenches to add another layer to the defensive system.

The French and German armies, to my knowledge, had similar systems. The details might have varied, but before the war, all major armies in the West had a similar approach to entrenchment and field fortification.

The goal in this system was to hold the fire trenches at all costs. I'll call this "forward defense." During artillery bombardments, troops would retire from the firing trenches and take shelter in the cover trenches. When the shelling stopped, soldiers would leave the cover trenches and move forward to man the firing trenches. There they would engage the enemy. If a section of the front line was weak or under heavy attack, it would be reinforced.

Forward defense was elastic to some extent, however. If a section of the firing trenches gave way or was overwhelmed, defending troops would withdraw and regroup, usually at a cover trench or a redoubt, and then launch a counterattack against the enemy with the object of regaining the lost section of fire trench. Despite this elasticity, the goal was always to hold onto the front line of firing trenches.

The British Army is my area of expertise, so I'll give you an example of forward defense from the First Battle of Ypres in 1914. The battle is complex and a bit of a mess to narrate, but for our purposes, we're jumping to the Germans' last real attempt to break the British positions around the Belgian town of Ypres on November 11, 1914. The British had constructed a good approximation of the system I described above. Despite the high water table and less than favorable terrain around Ypres (it's flat and very muddy in the fall), the British I Corps managed to build a front line of firing trenches reinforced with barbed wire and supported by a line of cover trenches and a series of redoubts (contemporaries called them "strong points" because they were a bit smaller than what prewar engineering manuals described).

On November 11, the Germans launched a concerted attack with the elite Prussian Guard. In many places they managed to break through the front line of firing and cover trenches, but British strong points contained the attack in the rear areas and inflicted heavy casualties on the advancing Germans, especially in places where the strong points were concealed in woods. British troops then regrouped, counterattacked, and threw the Germans back. Within a day, I Corps had recaptured many of its lost positions. Where it did not, the system contained the Germans well enough for a new line of firing trenches to be constructed, thereby preventing a breakthrough.

According to one staff officer in British I Corps, strong points "were the saving of the day. The [German] attackers blundered on them after they had broken through our line, and were taken in enfilade [i.e., the flank or the side] and broken up or driven into woods and hollows for shelter" (quoted in Military Operations 1914, 426).

When the Germans went on the strategic defensive on the Western Front in November 1914, they deployed forward defense, and a system similar to the one I just described persisted into the autumn of 1916. However, because the German army could settle in, it took this one step further. Redoubts and forward outposts became concrete pillboxes. Cover trenches evolved into deep, elaborate underground dugouts. Third and fourth lines of trenches were added in case the first lines gave way. But defensive theory continued to advocate holding the front line of firing trenches: forward positions were reinforced if weak, counterattacks were launched to regain lost positions, and abandoning ground, even temporarily, was generally frowned upon. For example, Erich von Falkenhayn, the Chief of the German General Staff, declared in November 1914 that forward positions "must be retained" since "even our relinquishment of totally useless objectives is considered a major accomplishment by the enemy and is then exploited accordingly" (Germany’s Western Front: Translations from the German Official History of the Great War Volume II, 165).

And this is how things remained, mostly (again, broad strokes), until after the battle of the Somme. See Part 2 below.