Did Viking warriors have any formal combat training? Were they organized soldiers, or simply pillaging farmers and fishermen?

This varies a lot depending on who the warrior is, and when we are talking about.

All of them would have some formal training, as every free man (a majority of the male population) were required by the county laws (the earlies written down laws are from the 1280s, but the tradition was much older than that) to own and train with arms, and we know that exercises in combat formations were carried out at the local thing at least yearly.

The hird, which was the personal guard of a great man or Jarl would be professional soldiers. These men trained since infancy to build strength and skill with weapons. Training included archery with progressively stronger bows, lifting and throwing stones and rocks, lifting and carrying logs and shield-pushing. The latter part included two rows of men picking up a shield each and forming a shield wall and then try to push through the opponents' shield wall in a sort of inverted tug-of-war. This trained dexterity, strength, stamina and co-ordination with the other man in the line.

The hird was the main way show of wealth and prestige to great man or Jarl in those times. To have a large, well-equipped and well-fed hird showed that you were a man to rely on, that took care of yours and had the might to defend yourself and those allied to you should the need arise. Men in the hird were thus often well-fed and thus often larger and stroner than most other men - indeed, it seems from the sagas that large and strong men often got offers to join a hird. The hird also trained in the shield wall and pig's snout formations. The former is a defensive formation, the latter an offensive formation in a wedge intended to drive through a shield wall formation.

Later during the viking era, a class of professional vikings, a kind of mercenary force, arose. Men that would join a protential viking for pay or promise of plunder. The Jomsvikings were such a force and like the hird they would be professional soldiers.

The larger viking hosts were almost always a majority free peasants rather than hird or professional vikings, and they were capable of organised and formation warfare. The typical way of fighting was to place the strongest and most well-armoured (usually hird or professionals) men in the first rank, wielding swords and axes, while the less well-armoured men wielded spears over their shoulders. All men threw spears, used bows and threw stones (or used staff slings) before the lines engaged. Hunting was free in pre- and medieval Scandinavia, and most men knew how to use a bow.

So, a bottom line. Even the free peasants had some training and organisation, but were usually put in the rear ranks to allow the more well-armoured and trained hird and professionals to keep the formation together.