I am preparing a Monte Cook's Cypher system campaign and would like to know how many skills a starting 1st-tier character would be trained in?
The answer varies based on some factors. The system does NOT provide any automatic levels of training which are universal to all characters. Instead, Training is found exclusively within abilities listed for each Character Type, Descriptor, and Focus. Not all of these offer Training, and some offer more than others. This is pretty balanced with other abilities, though.
Each of the four Types has an option among its starting abilities for a type of Skill to be considered Trained in. These are still chosen abilities, not automatic. For Example, a Warrior may choose "Physical Skills" as one of his starting abilities, which allows him to be Trained in two skills from the list provided in the ability description on page 25.
Some of the Descriptors provide Trained status in one or more areas. These seem to always be a specific area of focus related to the descriptor. For example Doomed (page 70) provides Trained Status in three different areas, perhaps to offset the terrible drawback of being Doomed.
Some of the Foci provide Training in certain areas related to the specific Focus. For example, the Focus "Conducts Weird Science" (page 109) offers the Tier One ability "Scientist," which lets the character be trained in one area of scientific knowledge of his choice.
Some areas of Training are for different situations rather than discrete abilities. This can create overlap. Any time Training can apply from two different abilities, the character will be considered Specialized. For example a "Warrior who Battles Robots" might be trained in Medium Weapons, and also Trained in fighting robots. (Not at Tier One, but this is just for an example.) If he fights a robot with a medium weapon, he is Specialized.
There are no extra bonuses for a third overlapping area of Training.