Unfortunately, material from different editions of D&D isn't compatible without a bit of legwork. Fortunately for you, the Elemental Evil Player's Companion has rules for playing a Goliath in 5th Edition.
To answer your question about minor actions:
A minor action is a small action, usually one that enables further action.
In 4th Edition, a turn consisted of three actions (minus free actions), a Standard Action, a Move Action, and a Minor Action. A minor action allows a character to do something small, say using an item, a skill, opening a door, or activating certain powers, such as marking an enemy. The rules for minor actions can be found on page 267 of the 4th Edition Player's Handbook.
In 5th Edition, a turn consists of an action (like a Standard Action from 4e), a move (Move Action from 4e), and a Bonus Action, if available. Bonus Actions are more like Standard Actions than Minor Actions, which leaves the Goliath power without a clear analogue. Thankfully, there are also Reactions (a small action available to you off turn), which is where the 5e Goliath Power lies. The rules for actions in 5e can be found on page 189 in the 5th Edition Player's Handbook.