When they say it sounds like the "G" in English, do they mean the hard G sound like Gary, the soft G like Genuine, or both?
It is hard, like in Gary. Hear for example the pronounce of га (https://ru.wiktionary.org/wiki/га#Произношение). It is the voiced counterpart of к.
In Russian there is no sound similar to G in Geniune. Such G is transliterated as ДЖ.
Sometimes, г is pronounced as the voiced counterpart of х. This is most common in Ukraine, and among people from Ukraine. You should not pronounce г like this, but you should recognize such г.
In word endings -ого, -его, and in the pronoun
его, г is pronounced like в. (But not in the words
Unfortunately, without context it's impossible to figure out what they, whoever they are, mean.
The English so-called hard G (or in phonologic terms voiced velar plosive) is in principle equivalent to Russian Г. However, its hardness in English isn't absolute since in certain positions it softens, retaining nevertheless its velar plosive character as in the words beGin, nagGing, rugGed.
Similarly, Russian Г can get soft like in the words друГие, ноГи, забеГе
The English so-called soft G can be transliterated into Russian as ДЖЬ, but Russian doesn't have such a sound, it's uncharacteristic of its phonetic system.
Instead, Russian has a sound denoted by the letter Ж used, in particular, at places where Г must become soft and so transform into Ж.
враГ - враЖеский
дороГой - дороЖе
слаГать - слоЖение
запряГать - запряЖён
моГу - моЖем
That said, the phoneme Ж itself is always hard in Russian.