As you are speaking of trenches, I assume you mean the Western Front.

The transportation of supplies in WWI, to include soldiers, is the primary responsibility of the Quartermaster Corps (Logistics Corps in modern warfare). There are two primary methods to move supplies. On a soldier's back or with transportation assets. The transportation assets available for resupply have evolved throughout the years, such as the addition of aerial delivery and the development of palletized shipments and specialized vehicles to quickly offload supplies, but no matter the advances in technology, supplies will always either be moved on transportation assets on on the backs of soldiers.

Short answer: Soldiers will move to the front lines either through trains, trucks, or marching or a combination of these three.

Long answer:

World War I on the Western Front is different from modern warfare in that on the western front, the lines of battle were relatively well known and rear security was not a huge concern, unlike in modern warfare where supply outposts are positioned close to the enemy lines, there is no clear "line of battle" and the enemy position is fluid.

A relatively stable battle line with means that more fixed assets can be put in place to move supplies. Units on the front built combat railways to move supplies from the division depot to the front, as roads were not always usable.

https://www.youtube.com/watch?v=vhlm67eLrpI

shows video of these combat railroads ferrying both soldiers and supplies to the front. As for vehicular transport to the front lines, MAJ Schoonmaker in his "The Supply Train -- Its Organization and Work" speaks of the general condition of roads near the front.

The worst roads are encountered very near the front, in an active sector. The artillery fire of the enemy say, within five kilometers of the front line is very destructive to the roads and in a great measure, prevents effective repair work. This is particularly true in the Somme sector, where the country is flat and practically treeless, and so without protection to the roads or the traffic moving on them. Most of the time, the enemy fire, is directed on the roads and batteries in the zone from two kilometers to four kilometers back of the line.

Here too, the entire area was swept with creeping barrage fire during the Allied offensive and advance. Roads were destroyed and there has been no opportunity to rebuild them in good shape. It is a fact therefore that even the main road is in very rough condition after one advances beyond the position of the French first line trenches before the offensive, —that is, in recaptured territory. Ambulances, for instance, are obliged to run largely in low gear from this point to and from the advanced dressing stations. The roads are passable but extremely rough. In this sector, too, upkeep of roads within about two kilometers of the front line is not attempted. Their position is too exposed and the shelling too severe. No vehicles approach the front beyond this limit, except the small "soup kitchens" and light supply wagons, and these only at night.

1LT Bernet describes in detail his travel from Le Havre to St. Nazarie and ultimately Rimaucourt via train.

At 7:00 A. M. the next day not realizing that one of the most important battles in the world's history was raging but a few hundred miles away, Headquarters Detachment, a portion of the Medical Detachment and Companies "A," "B," "C" and "D." marched to the railway station at Le Havre all leaving on the same train at 11:15 A. M. The departure was by a different sort of transportation from any that the men had ever known before but which has since become familiar to hundreds of thousands of American soldiers as "40 Hommes 8 Chevaux" (40 men or 8 horses) travel, because of the sign painted on the outside of each box car. About 32 men were loaded in each car, each being about half the size of the average freight car known in the States, and with them were loaded all their boxes of reserve rations.The wearisome train ride with the customary illogical delays, side tracking and switching which we have all come to associate with continental transportation began.

Although the route from Le Havre to St. Nazaire was a comparatively new one for members of the American Expeditionary Forces yet the children along the way had already become familiar with the generosity of the American soldier, and in little groups cried "Souvenir" and "biscuits" as the train rolled along, scrambling for what was thrown to them

As for the men the mere fact that a move was to take place and that that move was toward that ever—changing eastward line, was enough to make every man willing and anxious to bear whatever were the hardships of railway travel incident to it.

The travel across France was really full of hardships, although second and third class passenger coaches were provided for almost all the men. The trip to Rimaucourt was over a part of the country untouched by the ravages of war, and the only evidence that could be seen which indicated that a war was raging, were the train-loads of salvage, ammunition, and artillery pieces, the fiat-cars loaded with wrecked and burned aeroplanes, and here and there in the towns along the way a poilu on permission who showed by the lines in his face, his years under fire at the front. Now and then also a hospital train of wounded being rushed from the front to some base, passed.

Each detachment spent several days enroute at Is-Sur-Tille, the regulating station for that portion of the front, and then went on the short ride to Rimaucourt.

The general poor condition of roads near the front does not mean that trucks were useless for bringing soldiers to the fight. MAJ Schoonmaker says again of trucks and the movement of soldiers:

It was with a good deal of hesitation that the General-in-Chief, at the beginning of the war, authorized the transport of troops by automobile. Now he has changed his ideas, and to date we transport about 500 battalions per month. It therefore goes without saying that the use of trucks for this work is very important.

1LT Bernet describes a portion of his unit moving to the front via a convoy:

On July 21st on orders received from Motor Transport Headquarters at St. Nazaire, Lieut. Pierson and a detail of 30 men from the detachment there started out convoying 22 G. M. C. ambulances to the Zone of the Advance, with orders to report at Dijon for further instructions. That term "Zone of the Advance" had a magnetic sound to it then. As the detail clambered into trucks to be taken down to the park to pick up their convoy, every one looked after them enviously for it seemed they were among the chosen few who were starting off to the promised land, the. land of adventure.

Even though the condition of the roads was bad near the front, the ability for the US Quartermaster Corps to use the French Automobile Service freed up freight capacity and allowed greater flexibility for movement. The total mileage of roads greatly outnumbered the total mileage of rail.

For shorter distances, just as in modern warfare, marching was the quickest, most efficient method.

For a movement under 25 kilometers, there is really less fatigue for the troops to march the distance than to march to the point of embarkation and from the point of debarkation. In this case, however, the truck will help, for it can go and get the infantry packs, follow a different route and turn them over to the troops at destination. A Group will carry the packs of the entire Division.

(1) United States. Army. Quartermaster Corps. (1917). Manual for the Quartermaster corps: United States army. 1916. Washington: Govt. print. off..

(2) Schoonmaker, O. [from old catalog]., June, W. D.. (1918). The supply train: its organization and work. N[ew] Y[ork] city: E.N. Appleton.

(3) Bernet, M. Emil. (1919). The Three hundred and fourteenth motor supply train in the world war: an account of the operations of the supply train of the 89th division from its organization until its demobilization, including maps and complete rosters and appendices. [St. Louis.

(4) Tables of Organization, United States Army 1917

(5) ADRP 4-0 Sustainment