"I got" is the past tense of "I get":
"I got a dog yesterday."
What you might be thinking of is "I've got" which is the contracted form of "I have got":
"I've got a pencil."
"I have a pencil."
These two can be used interchangeably. But you will find that using "have" can sound a little more formal and "I've got" a little more casual. Certain expressions just sound better using the more casual of the two:
"You think you're better than me? I've got $100 that say you're not."
However, the difference is very minimal and you can usually get away with using either.
In regards to "they" and "them"; "they" is the 'subjective' form of the pronoun and is used when the pronoun is doing the action:
"They skate everyday."
"Them" is the 'objective' form of the pronoun and is used when an action is done TO the pronoun.
"We greeted them with cheers."
The easiest way to remember is:
THEY did it to THEM.