The other answer already explained the different usages of 「下去」,
I would look into the logic behind it, and illustrate it with some examples.
「下去」is basically [下- down / downward (adverb) + 去 (directional particle)] structure that modify a verb before it.
- 踢下去 kick down
- 咬下去 bite down
- 劈下去 hack down
In the three examples above, the role of 「下去 」is simple - it indicates which direction the verb apply to (*downward)
The other use of "V+下去" indicate continuation of an action 「下去」= ( down the order/ continue next)
- 讀下去 (keep on reading)
- 唱下去 (keep on singing)
- 想下去 (keep on thinking)
It functions like "繼續+V"
- 繼續讀(keep on reading)
- 繼續唱(keep on singing)
- 繼續想(keep on thinking)
p.s. I just realized I may actually be confused by 接 in this sentence... Possibly could someone explain it to me? :/
接 here is a verb "to continue"
"父亲**接 下去 *说:.."
"my father **continued, *and said:..."
- "and" in English also indicates "down the order in time line". For example:" catch and release" = "catch first then release right after"