The basic difference between a regular 〜て and a 〜てから is emphasis. てから emphasises the order things go in.
〜てから is used much much more rarely than a regular て for joining sentences. I'd estimate that for every てから, you'll see fifty sentences joined with a plain て.  When I see 〜てから used, it's immediately a flag for me that I should pay attention.
Here's a sample sentence from the Tanaka corpus:
After Grandma's sudden death, Grandpa began to age rapidly.
If written without から, it would read:
Grandma died suddenly, and Grandpa aged rapidly.
Doesn't make much sense, does it?
Another sentence, from an old JLPT test:
Mariko-san started basketball after she went into high school.
Mariko-san went into high school and started basketball.
There's nothing wrong with either sentence, but in the former sentence, you know that she didn't dabble in basketball before entering.
In your directions example, the 〜てから version would (exaggeratedly) read something like this:
Only after turning right, go straight. After going straight, turn
left, and only after that, go straight.
Often, when used, it makes a big difference in how the sentence reads. So the rare times you run across it, pay attention to it.
 I'm not really sure why they're teaching it at such low levels since it's that rare.